Stories like these are why my boys aren't allowed to play travel ball. Regular little league season and allstars if they make it. High schoolers needing tommy john and shoulder reconstruction is absurd.
In my case, playing lots of games was not really the issue. Catching nearly every game that I didn't pitch was the issue. Those two positions involve the most throwing, require drastically different throwing motions, and often require near-max effort throws. I was incredibly competitive and didn't hurt, so I didn't know any better.
The other major factor was going year-round. If I wasn't playing, I was training. The notion of taking two weeks or a month off from throwing was foreign. The arm can withstand a lot of punishment, but it has to have some time to rest and recuperate.
By my third year in college, all the wear and tear and minor injuries had weakened the arm enough that the shoulder gave out during an awkward rep while bench pressing (low weight reps, not maxing). The UCL in my elbow had enough partial tears that it eventually ruptured during the latter stages of the rehab throwing program.
If I knew then what I know now, I would have significantly altered my summer ball and training routine as a kid and teen. The way my back, hips, knees, and feet/ankles hurt now, I would have only caught one or two games a week instead of four or more. I would have used the games to focus on my offensive game and found a way to play a less demanding position most of the time.
That said, I am still a strong proponent of utilizing travel ball to develop one's game. If your kids are not pitchers and/or catchers, they very likely will never run into the same problems I did even if you do let them play.