Fame
You've probably heard about celebrities' fashion stylists who've gotten so much buzz they get their own fan bases. You couldn't possibly have missed the high-profile lawyers who repeatedly get the same misbehaving actors out of trouble on both coasts. You might wonder if a professional team's physician looks for that same kind of publicity.
Generally...no. They don't. At the most fundamental level, someone becomes a physician because he or she wants to help people, to ease suffering, and to help make advances in disease treatments. Not because they want to be famous.
Looking specifically at sports physicians, remember they work long hours and many maintain their own professional practices. On the college level, and especially the pro sports level, team physicians travel a lot, living out of suitcases and diagnosing and treating multiple player problems at once. If they wanted their fifteen minutes of fame, there are probably easier ways to get there.
Of course, it's possible a sports physician might get some press by keeping an injury-prone athlete, say, an Olympic competitor or pro sports superstar, healthy enough to compete year after year. In that case, the sports doc might get a mention at the sportswriters' banquet. He might also get a few new clients from all the publicity.