My older son plays competitive lacrosse. He also plays football, and is on a fun flag football team with his buddies. His comp team coach has always said 'I love that kids play multiple sports' etc. etc. I've also heard this from his football coach and other coaches too. They ALL say that.
And BTW I believe that they're sincere in that belief and want to avoid specialization. It sounds great in theory.
But in practice...those same coaches get annoyed and punish your kid when they can't make it to every training, camp, scrimmage, practice that is recommended because they conflict with the other sports. So in theory its wonderful to say 'we want multi-sport athletes' but in practice you simply reach an age (and it's way earlier than it should be) where your kid cannot make competitive teams and play multiple sports. There aren't enough evenings in the week to do it, and their skill doesn't increase as rapidly as it should. So in practice, this is why kids specialize sooner than they should: their schedules force them to. Eventually you have to choose between a rec lacrosse league and the trainings the comp lacrosse league is sponsoring that fall on the same nights, or between football tryouts and the lacrosse tournaments, etc. etc.
NOTE: Completely ignore what I said if your kids is one of the best players on a team. Coaches don't care what that kid does as long as he shows up for games and dominates, but for everyone else who's fighting for a spot, what I said is true.