A lot of time with team physician stuff is after clinic hours - 8:00-4:00 seeing patients, 4:00-7:00 seeing athletes vs going to games on weekends. So plenty of time to see the patient load, but you lose a lot of nights and weekends.
No one is forcing anyone to do sports medicine or to be that level team physician. 95+% of people do it because they really love sports and would be attending sporting events in their free time anyway, enjoy treating high level athletes, enjoy helping these people achieve accomplishments that only 0.000001% of the population could do, and enjoy the perks of the job that are not in cash compensation - i.e. relationships with politicians, business leaders, athletic directors, famous coaches/athletes, etc. which in and of themselves are worth something.
We billed for what we did (99212/3/4 training room visits and office visits, injections/procedures, PRP, surgery, etc.), but no, we don't bill per hour like a lawyer would so overall your time is largely uncompensated compared to clinic time. In certain academic settings your "admin time" or "education time" is utilized for this game/team coverage so that you don't have to worry as much about production, but my set up was private practice so we don't have a health system/government slush fund to keep us afloat.
Sports are entertainment, but can bring great joy to cities, communities, and all the behind the scenes employees at these organizations. I feel that communal benefit is worth some amount of economic benefit to those producing the product. If you don't think high level or professional sports are worth the cost then you do not have to consume the product and that is perfectly fine. More importantly direct your angst at the media companies that have turned sports into the monster it is today. The coaches and players are trying to get their small slice of the pie while their knees and shoulders still work or until they have a losing season. I don't blame them for utilizing their skills while they are in demand. I'd do the same thing if my mind/hands only let me be a physician for 5-10 years.
I'm now doing 30% sports/70% spine private practice, help with some local high schools for fun, and have a lot more free time to enjoy with my family. I enjoyed what I did, personally know a lot of power players in D1/pro athletics (which has perks for tickets, they are generally cool people, etc.), have some great memories/experiences from it, and now have a loyal patient base that I otherwise would have taken 3-4x as long to build.
I'm satisfied with the above ... but some may not or think I'm a fool. That's fine. I've played my career game successfully and have been fulfilled so far.