Always a great topic...
The son who ended up playing college ball liked doing WAY too many things. At the end of the day, we supported doing what you have passion for, exposing our kids to lots of different activities and interests, having fun being kids, balance, etc. Boy, did he take advantage.... But better to have a life too full with no regrets than a life too narrow with too many "i wonder if.." or " i shoulda.."
That said, had he been throwing high 80's as an underclassman with high projectability, it would have been much more difficult, I'm sure, to maintain that approach. The offset, of course, is to remember that baseball can end at any time for a variety of reasons. You don't want your son to be left not otherwise having had a fulfilling and well rounded life as a HS kid.
There are certainly sports and activities that are generally higher risk than others... we lose a few in our HS program each year consistently to either football, snowboarding or skateboarding. Other common season-ending injuires come from wakeboarding, bmx, dirtbiking and rodeo. Still, if those sports are what they really love doing...
Injuries are going to be part of the equation, no matter how big the bubble. 3and2 makes a very good point about being over-pitched as probably the biggest concern you will have to watch out for.
If he is a real prospect, those comments are going to come... mostly from baseball people who's primary interest is to see him succeed in baseball (edit to add - it sounds like in this case, a pitching coach more interested in seeing him succeed in pitching). There is some merit and warrants consideration but keep in mind that they are less concerned to see him succeed in life in general as you.
Educate them. Help them make reasonably smart decisions and choices. But let them live.