I am not sure but if I remember correctly the NFL teams play 15-16 games during regular season and a couple more in the play-offs if they can make it the whole way. Why such a short season, in comparison to other leagues such as NBA where teams play 82 games in regular season and playoffs are usually a few games each round (best of 5-7-7-7)?
The physicality of the NFL is much greater than those of the NBA and MLB for example. Football is one of the most strenuous sports, and is much more of a full contact sport. When you are coming into physical contact as much as NFL players do, then you are more likely to fatigue and be at a higher risk of injury.
Players need a break to rehab from injuries, get ready for the off-season, training camp, and the preseason. The human body can only take so much, especially and when you practice several days a week, and play full contact once a week.
In the NBA for example, players do not have to worry about being tackled by a 300+ pound lineman from play to play. It is illegal to tackle an opponent in the NBA....that's not to say there's no physical contact. However, compared to the NFL and the sport of football, the NBA and the sport of basketball have much less physical contact.
Within the sport of football, a kicker has a longer lifespan than a runningback.
A lot of the reason has to do with tradition. Football has always been a once a week event sport, and hockey, baseball and basketball have always been every night grind type sports.
There isn't really a reason that football games couldn't be played twice a week, these guys are supposed to be world class athletes. Yes the game would change, and yes the quality of play would diminish, but only until the league and the athletes in the league adjusted. But the NFL and college games have built up as a once a week event and as such they have developed their revenue model around it being a once a week sporting event (baseball, basketball and hockey teams don't build 70-100k fan arenas anymore because they can't fill them every night, football teams do because they can fill them once a week).
I don't feel like there is a physical reason that the NFL could not play games more regularly, rather, the way they have built up their brand, and their infrastructure, they need to maintain the relative scarcity of their games. It is no coincidence that the once a week, 16 game schedule league has every single game broadcast on TV, not by home team announcers, but by national, professional broadcasters. It's no coincidence that many of the teams in the NFL can rather easily fill a 70,000 seat arena. They have built their brands around the once a week event, and adding any more games (especially a second game a week) would only serve to dilute the brand and infrastructure that they have built.
In contrast, MLB, the NBA and the NHL have managed to do quite well with more local coverage of their teams, with more games, and with infrastructure that accommodates their more protracted schedules (In fact infrastructure and broadcasting contracts is a major reason that the NBA is not considering permanently shortening their schedule to the 62 game season that they played this past year). The fact is that all of these sports would have a potentially better products if they played fewer games. Imagine if your local baseball team played every 5 days and your ace got to throw every game. Imagine if basketball teams got to have fresh legs for every game of the season. Imagine if your local hockey club was able to shorten their lines to only 2 shifts, because they only played once or twice a week. (the argument against doing this would come strongly from baseball and basketball in that both are arguably rhythm sports where if you don't practice regularly you could potentially lose your hitting or shooting touch)
In conclusion, Football plays 16 games because of history. That history has led them to develop their brand and infrastructure (both physical, and contractual) around a scarce product(16 game season), rather than a plentiful one (162 game season).
I think the basic reason gridiron teams play so few games is the nature of the surface on which they play. The artificial surfaces they use in many stadiums today are extremely high-friction. This friction is what causes sufficient wear to prevent NFL teams from playing more than one game a week or for more than four months. Natural grass, though not as bad as artificial turf, is also a fairly high-wear surface especially in warm weather. This would preclude starting an NFL season during mid-summer even if it were more feasible for broadcasters and other stakeholder.
Although ice hockey is an extreme contract sport like gridiron, ice is a low-friction surface that exerts very small forces on players’ bodies even during violent contact (of which ice hockey has plenty). As a result, if they do not suffer a collision with a puck or sharp skate blade or stick (which in fact can be more dangerous than collisions with giant gridiron linemen), ice hockey players can keep going for a long time at a physical peak.
European soccer surfaces (generally natural grass) do cause wear on bodies but it is limited by the cool and damp climate and youthful soils. Australian football (AFL) is played on much drier surfaces owing to the hot and dry Australian climate and extremely ancient (formed in the Carboniferous vis-à-vis Northern Hemisphere soils from recent glacial till or mountain alluvium) soils that require much denser rooting systems for minimal nutrition. As a result, even a 22-game season with much less physical contact than gridiron is physically very difficult for tall and slimly built people.
There is the argument that because football is very physical on the players, having the season much longer is not reasonable. That's part of the reason.
They could obviously work around this by playing less often but over a longer period of time, but that would confuse the schedule (which is a strength they have over other sports). Or they could sign more players, which would increase costs, but television revenue and ticket sales are good, so the problem might rather be that there aren't that many more good players available.
They can't start the season much earlier. The NFL relies heavily on integrating college drafts into the teams, and the college players aren't going to be fully available until they have concluded their studies around April. Many NFL teams have in the past shared stadiums with baseball teams (not so much anymore), and so adding more games before October would create scheduling difficulties. And television networks don't want to show high-value programming during the slow summer months.
They can't end the season much later. First of all, you don't want to move the Super Bowl from its established location. Also, football is traditionally an outdoor sport that's at home in the north, and the weather dictates that you shouldn't play too many games between January and March. They do play the occasional playoff game in the snow, but you don't want that every week. Then, in February, you have the Winter Olympic Games to contend with every four years, and in March baseball season gets slowly going.
So all things considered, the current setup is the sweet spot that works for the sport and the league. They could probably expand it, but only a bit without diluting the product.
Football at the NFL level, and even at the college level, is such a physically-demanding sport that players require at least one week of rest between games. Beside the typical injuries seen in any sport—muscle strains, sprains, bruises, etc.—pro football players are increasingly having to deal with concussions (i.e. "brain bruises") because it's not just a contact sport, it's a collision sport. You don't see boxers fight every day, or every week, for the same reasons: the sport is so brutal that the players need extra time to heal from the previous contest.
On the other hand, the shortness of the NFL season has an extra benefit: it makes each game much more valuable in terms of determining a champion. A short season increases the drama, and since sports is fundamentally entertainment, that can be a good thing.