Some questions.
Probably in all sports suffer to some degree from a rash of overuse injuries. But does any sport suffer more from it than baseball -- pitchers in particular? I'm guessing not. Maybe tennis? Swimmers?
Some sports you couldn't possibly play all year round -- you would be far too beat up at the end of the year. I'm thinking football, mainly. Or things like it -- rugby, possibly lacross. So presumably it's mostly non-contact sports that have morphed into being all year round sports.
Is the main reason to be a multisports athlete to prevent overuse and burnout? Or does playing multiple sports actually somehow help you play each one better? I'm not sure, but I suspect the former not the latter. It's not as though the skills you need to be an outstanding point guard have much to do with the skills you need to hit a 90+ mph curveball. Ask Michael Jordon about that.
Football and hockey are probably the two sports where you would be least likely to survive playing "year round." That's one reason football has developed the 7v7 that they have. Baseball/softball, swimming, soccer, distance running, basketball, and volleyball tend to be the worst for us. Our kids play all the time. I think volleyball tryouts for club teams fall only two weeks after the school season ends. That will run all the way until the middle of the summer until they start the school season again, basically.
I have to have conversations with kids from a lot of different sports on a regular basis about the need to back off. We have to help them prioritize what is truly important and what can be missed. Sometimes that means I even have to go as far as to put a kid with a lower body injury on crutches or a kid with a shoulder issue into a sling; just to get them to STOP.