Most "head team physicians" among the NFL, NBA, MLB, and NHL are orthopaedic surgery trained. However, every one of these teams have non-operative physicians who are a part of the medical staff as well. So, there is someone to take care of the ortho/operative side and someone to take care of the medical/non-operative ortho side. Also, among other sports such as minor league professional teams, boxing/combat sports, and Olympic sports, "head team physicians" have wide ranging training backgrounds.
That said, here's my typical run down of pathways to sports med:
1. Orthopaedic surgery: very competitive, need great board scores. HAVE to like surgery. Residency is grueling but practice, depending on setting, can be very busy to cush-ish. Fast track to "head team physician" for major pro teams if that's your thing. Not as much training in non-operative MSK medicine compared to PM&R. Great pay all around. Only orthopaedic surgery trained physicians can do orthopaedic surgery sports medicine fellowships.
2. PM&R/physiatry: generally less competitive but getting a lot more competitive especially at the "top" places. Have to get over the fact that when you say you are PM&R/a physiatrist most people are like "huh"? (this can actually be a hang up for people). Otherwise have to like a lot of different areas: neuro, ortho, a little bit of medicine. If it's your thing, out all the other paths to sports med docdom PM&R is second to ortho in terms of procedures available (interventional spine, US injections, EMGs). AWESOME lifestyle. Actually great pay when you consider the hours/relative lack of stress involved. PM&R trained physicians are eligible for PM&R based ACGME fellowships (for which only PM&R physicians can do) and primary care/family medicine based fellowships.
3. FM: competitive to not competitive at all: can literally walk in to some of the lower tiered FM programs (but they may not have any sports/MSK faculty). Great jumping off point for the medical care of the athlete since your backbone is the medical care of all ages and types. Much less neuro compared to PM&R and much less ortho compared to ortho/PM&R. But you will be THE medicine guru when it comes to athletes. Ok lifestyle depending on your practice mix. Pay is moderate.
4. IM: competitive to not competitive at all. Have a medicine backbone like FM but get MUCH less outpatient exposure these days in residency in most places. Even at some of the top IM residencies in the country, IM residents come out with the MSK knowledge of medical student. So IM residents interested in sports really have to push for outpatient MSK/sports exposure since most programs aren't geared that way. Plus as mentioned IM residents never really take care of kids which is a minus for a well rounded sports doc. Lifestyle can be what you make it. Pay can actually be pretty high if you take hospitalist type jobs on the side.
5. Peds: competitive to not competitive at all. Have an excellent pediatric medicine backbone and good training in outpatient medicine, but in general, peds residents are relatively deficient in adult sports medicine when coming out of residency compared to all other specialties that feed into sports med. Lifestyle can be what you make it depending if you have a strictly outpatient practice vs. urgent care vs. part-time hospitalist. Pay varies from low to high depending on your practice set up.
5. EM: competitive to moderately competitive...in general you have to actually be a decent candidate to match into any EM program. AWESOME at taking care of acute issues/injuries medical and MSK related, not so awesome at longitudinally taking care of patients (or at least not part of the core residency training at most places). Lifestyle and pay is what you make it depending on how many ER shifts you take on the side.