The problem with having football Feb.-May is other sports. Most teams only have one field, so you run into playoff soccer, lax, flag football, and a lot of schools track teams practice field events on football field. You move them and you have same problem just with different sports. Also, track, lax, and baseball guys who play football as well. You would have to change a lot to move football.
On top of that, I would add that not all HS football players are healthy or without underlying conditions. Some players do have asthma, diabetes, obesity, undiagnosed heart issues that slip by etc. You are playing roulette with the odds. While most kids won't die or will be asymptomatic I would hate to be the guy who gave the go ahead and 1 kid dies or the coach of that team.
I would be interesting to have football season during February to possibly early May. I have always felt that this would be a great idea with no Hurricanes, very little severe thunderstorms, and less heat than August and September. I wonder if Rutgers and Princeton would have played that game in March instead of November in 1869, maybe football would be a spring instead of a fall sport.