Imagine a scenario: You get hired by a big SEC football program as a consultant to come up with some new ideas for long-term team & player development strategies. On day one you walk into the head coach's office, set up a projector, open Keynote, and present your Four Point Plan:
1) Send most of your team to randomly selected countries for two years.
2) Make sure they play no real football during that time. If they do play at all, it is critical that they only play two-hand-touch once a week with non-athletic kids who all get a turn being quarterback.
3) Depending on their assigned country, feed them too much or not enough.
4) Instruct them to talk to strangers about Jesus for 12 hours a day.
You close your laptop, sit back smugly in your office chair, and look at the coach. He calls you a traitor, pulls a gun, and tells you to get out of his office before he shoots (it's the SEC; they're intense like that).
Far from being the unequivocally stupid idea you almost got shot over, however, the 2-year missionary service that most BYU athletes undertake has actually been considered by some to be an unfair advantage. For better or worse, BYU coaches don't have the option of rejecting this proposal (and they wouldn't last long in Provo if they tried). So, then, is it a blessing or a curse? In order to decide whether it's a boost or a bummer, I've done an extremely biased, off-the-cuff, unscientific research project only moderately supported by data. This will definitively settle the issue for time and all eternity. Let's begin.
Thinking of real life as a video game (which I often do), we can put a number value on the key player attributes. For our purposes we will use four concrete traits: Speed, Strength, Skill, and Intelligence, and then one fluffy vague one we'll call "Fire" (this could be substituted for any other motivational-speech-type noun that your coach always said is the most important, i.e. "Heart", "Desire", "Will").
Anyways, the question then becomes simple: will the numbers for those 5 attributes be higher or lower on average for a player who has served a 2-year mission?
Your coach lied; this is clearly not the most important attribute unless you want an inspiring movie made about you. The short, slow, awkwardly enthusiastic kid that is clearly a 99/100 in this trait only gets in in the fourth quarter for a few plays and then runs around with too much energy like a Labrador that has been kept inside a Manhattan apartment for too long. Any coach would trade that enthusiastic kid in a second for someone who can bench press twice his weight.
Speed is the most important attribute in question. If you had an entire team of dumb skinny guys that could run 4.2 forties, you'd probably win a ton of ball games. Given the chance to turn one knob up to 11, you would choose speed on pretty much any player besides the linemen and quarterback (unless your quarterback is running like Taysom).
In a research study that surveyed 646 running careers and over 5,000 individual performances, the peak performance age was 26 (ranging from 23 ½ for some shorter distances to 31 ½ for marathons). The fastest guys in the world all had their peaks long after they would have graduated from college. Michael Johnson was won gold medals from ages 25 - 30. His all-time personal bests were set when he was 27 (100m), 29 (200m), 33 (300m), and 32 (400m). Carl Lewis's fastest ever 100m was when he was 30, and he didn't win any golds till age 23. Donovan Bailey has a similar story. The only personal best of Usain Bolt's that was in normal college football age was his 400m at the age of 21, and nobody runs that far in one play except for this guy.
With the NCAA eligibility requirements, it's safe to assume that the only people who are getting close to that peak age for speed are those who get time off that doesn't count against them (i.e. return missionaries).
For the slow guys, strength is the most important attribute. These are your linemen, linebackers, and blocking backs (though it never hurts to have your safety benching 380 either). For this research, I scanned world records for weightlifting across different events and weight classes. Here is the distribution of ages when the records were set:
20, 20, 22, 22, 22, 23, 24, 24, 24, 24, 25, 25, 26, 26, 27, 28, 28, 29, 30
That gives us an average age of 24.7 for world-record-setting. We'll ignore the idea of conditioning until later, and go ahead and say that 22 is better than 20 when it comes to strength (as long as you come home without a parasite that sucks all the nutrients out of your lunch and give you tons more baby parasites in return, in which case it would be harder to build 250 pounds of muscle).
 According to the data, it's impossible to set a world record for weightlifting at the age of 21. That's not me talking; that's science.
 In my mission (Paraguay), it was pretty much a rite of passage to get worms. Everyone had them at least once. I got one little worm friend who put a toxin in my blood that degraded a part of my optic nerve and gave me a condition called palinopsia that makes me have double vision in low light conditions for the rest of my life. Fun!
This one's a bit more nuanced. On the one hand, coordination tends to mature along with your other athletic abilities (look at the prime ages for soccer players, basketball players, etc). On the other hand, a lot of skill depends on things like muscle memory, experience, pattern-recognition, and judgment: all the "10,000 hour rule" things that unfortunately take 10,000 hours to develop. Many of those subtle things are probably best developed with continuity and likely get disrupted by 2 years of singing hymns, teaching gospel principles, and telling endlessly dumb jokes that get passed from missionary to missionary like a persistent virus. So a quarterback loses some touch in his throw, a halfback loses feel for when to zig and when to zag, and the pass block gets a little sloppy on the line.
On the other other hand, they say mental repetitions are as good as physical ones, so if you had a super motivated receiver who visualized catching passes for two years straight between knocking doors, he might come back better at it than before. Too bad this has never happened because imagining fake passes gets boring after about one minute.
ADVANTAGE: EVERYONE ELSE
Don't trust the stereotype of the big dumb football player. Some of those guys are wicked smart. They might not know how to spell or multiply 2-digit numbers, but that's just because they skipped the last period every day in high school to go run wind sprints in a harness hooking them to a sled with a fat o-line coach standing on it eating Doritos Locos Tacos while you nodded off in the back of the class and drooled on your desk before going home to play Golden Eye on your N64.
Which one's harder: solving for x by cross multiplying some fractions or standing in the backfield, tracking five possible receivers, watching the quarterback's facial expression from 20 yards out, and trying to simultaneously analyze everybody's trajectory, openness, and speed while cross-referencing the field position, game situation, weather conditions, and the tendencies of the quarterback? When you think about it that way, shouldn't there be a Nobel Prize category for free safeties?
In terms of the mission effect, we run into the same issue as with the Skill analysis. Older is better in your 20s when it comes to peak intelligence and problem-solving skills (the same research article that found peak age for sprinters measured peak age for chess players, which came it at a whopping 31), but continuity is again a big factor: it's hard to get "football smarts" without playing football.
A lot of those "football smarts" can be enhanced by emotional maturity, though, which is probably the single biggest development in return missionaries. Combine that with the fact that 19-year-old dudes are without a doubt the dumbest humans on the planet, and I think the two years of growing up at least counterbalances the two years of not seeing football.
ADVANTAGE: NOBODY (TIE)
 He's double fisting both the Cool Ranch and Nacho Cheese flavors. Also, this adds about 8 ounces to the weight of the sled.
 This was the first time chess players have ever officially been lumped together with athletes, which was a huge vindication for them. They will now be joining the jock's table at lunch in high schools all across the country. However, now that Magic: The Gathering tournaments have been featured on ESPN, the meaning of "athlete" has fallen way too far to ever be recovered. Now nobody knows who's supposed to pick on whom, which has become one of the more serious existential questions facing today's youth.
 I'm surprised we didn't evolve to just hibernate from age 16 to 22. I feel like that would have been a really great development for the whole species.
As much as I made fun of these kids in high school, there is definitely a benefit to the kind of irrational emotional energy that you see in many dedicated athletes. Certain positions (but mostly middle linebacker) have a lot to gain from having an insane and delusional individual who has a deep-seated desire to break other people physically, emotionally, and spiritually in order to stop them from carrying a leather ball to the other end of the playground.
I just can't imagine that doing anything where you're regularly singing "Scatter Sunshine" and "Love One Another" is going to help much here.
ADVANTAGE: EVERYONE ELSE
Now that we've explored how the mission affects individual players, we need to examine the cumulative effect that having most of your team leave for two years has on a team and a football program as a whole.
For this part of my research, I sought the opinions of an authoritative expert on the subject: Matt Bauman. Matt was a walk-on who had about a thousand tackles a game as a freshman middle linebacker, served a mission, then returned as a dominant force on the defense and eventual team captain. If anyone has a good perspective on how missions impact the team, it's Matt.
"Coaches spend so much time on discipline at other schools: making sure kids stay out of jail, go to class, get good enough grades, that stuff. With most of the return missionaries, their academics are better, they're a little more mature, so the coaches don't have to worry about that as much and can think about more important things."
"Having people grow up a little bit really helps the team's mental game. In football there's so much to understand and so many different things to remember. I feel like the return missionaries were more focused, so they were able to learn everything better. Plus there are less distractions and less problems off the field, so they have more time to think about football."
"The shared experience of the mission is really a big thing. Everybody went through a lot of the same things, and that's a commonality that you can build off. On the mission you really learn how to interact with people and deal with a lot of different situations, so having a team full of guys who learned all that helps a lot in terms of cohesiveness and leadership."
 I couldn't take notes fast enough in our conversation for these to be 100% accurate quotes, but they're pretty close. For example, I don't think Matt used the word "cohesiveness" but I can't read what I wrote; the word I wrote looks like "curvnetck," which I can't find anywhere in the dictionary, so I took some liberties as a translator.
"The conditioning coaches give you some exercise bands to take on your mission, but that's pretty much the whole program. There were a few guys who came back ripped, but that's definitely the exception. I think Vakapuna came back more ripped than when he left. For most people, though, they come back in such bad shape that it takes about a year to get back to the level they were at in high school. So some of the team are in their junior year before they really are back to where they would have been.
I tried to work out every day, and it still took about 8 months before I felt like I was in good shape again. Some guys, though, just look terrible when they come home. And then a lot of them get injured too because they're trying to bounce back too fast.
I think for the coaches, that's really hard to manage when so many of your players are just trying to catch back up to where they should be, and you always have some of them because everybody gets home at different times."
"The timing of players coming in and out really gets thrown off. A lot of players are forced to red shirt because they're leaving on missions mid-season. There's really only one short window in January where you won't leave or come back in the middle of a season, and there's obviously no way to control that. So the coaches are dealing with weird scholarship situations, weird recruiting arrangements, things like that. Plus you never know if someone that's good before will still be good after, and how soon they'll be back in shape, so it's hard to plan for. I know there are some kids who were good when they left that sucked when they came home. I can't think of any specifics, because it's hard to remember the people that weren't good, but it definitely happens."
After a careful review of the evidence, I have concluded that the mission effect is a net positive for BYU. Having players with more potential to be bigger, stronger, and faster is good. Having players with less potential to be idiots is good. Having a team that understands each other, communicates better, and doesn't have as many problems off the field is good. Having a bunch of fat kids who were overfed in California or skinny kids with tapeworms from South America is bad, but anyone who has the discipline to wake up at 6:30 and try to get strangers to let them into their houses to talk about God all day every day for 2 years can certainly muster up the energy to spend a few hours in the weight room and drag a coach around the field on a sled for a while.
It might take them a while to get back in the game, but once they do they'll be able to face down the 19 year old halfbacks across the line from them who are still winding down from puberty, completely murder them on a blitz, and then with hope love and charity, help them back up, tell them good game, and then ask them if they'd like to get together after the game and hear a message about how families can be forever.
 This is the definitive opinion on the subject and, like a Supreme Court ruling, will now be the precedent for deciding any and all arguments about the subject.