We got to know the HS basketball pretty well - he encouraged and hoped his players would play football in the fall. Did kids get injured - yep, sh*t happens, but beyond the different muscles used - other sports create/cause certain team dynamics and reliance to allow players to realize they're not "it" at everything. A few may be, but it's hard to be the best at 3 sports unless you play all 3 year round which as most parents probably realize means you have no time for anything else. In the long run it's about building up a certain mental toughness that coaches like. Future employers like former athletes too - personally driven and understanding of team dynamics. I think the NCAA runs a commercial series on that - some find it annoying, but there is truth to it! ;-)...
When my kids grew up we "competed" against the s sport in the spring for baseball players. Drove me nuts to see them require 8-12 year olds to play 3 seasons in order to stay on some travel team. I feel they were using the "status" of travel for some parents minds as the primary motivator. When we started Fall baseball I would tell kids/parents - the s sport is the in season sport, choose that and then be a fill-in at baseball. That is if you're not playing s, then come to the baseball field - we'll treat it like a pickup game and put your kid out there. We ran our rosters with 10 players (small town).
fwiw: Son#1 did play hs football, he did get injured (broken jaw) in Soph year, but still continued to play football & baseball. He gave up hoops after 1 season (hard to get a spot after missing soph season - he tried though)... He switched to the other side and was one of the leaders in the student section... His Sr year for baseball a new coach made it clear that he was going to "play" the Jr's... None of the Sr's came out and my son got into a bit of trouble writing a letter to the editor detailing why the group of 8 Sr's chose not to play. In college, he played 1 year of football at a D3 high academic, ended up being the treasurer, then president of the baseball/football fraternity, and a leader in the greek council. In my mind, all his sports activities helped shape him and understand interpersonal dynamics that one may not see playing just 1 sport and are considered the best.