Who pays for athletes' heath care?

Factual Questions
1

Who pays for the health care of professional athletes? Their teams? Their unions? Themselves?

Let’s say a 20-year-old pitcher who’s not considered a very promising prospect gets injured in such a way that his only shot at continuing his baseball career is “Tommy John” surgery. Can the team (or whatever the answer to question 1 is) opt to cut its losses and figure that his career is unlikely to justify the cost of complex surgery and rehab? Or are they under some sort of obligation to underwrite any medical procedures that could continue his career in baseball?

2

In baseball, the team pays.

3

I would expect that this is written into their contract, and is probably paid by their employer.

I doubt very much that such cost/benefit analyses as you mention in the OP would be done - medical insurance typically either covers a procedure or doesn’t, regardless of whether a person’s career or potential income would “justify” it or not.