Professional sport leagues in the US do share revenue...and if you are looking at things inside the league, the gains are "socialized". I'm a big fan of this model as opposed to a league with unconstrained finances...say like the EPL (which I still enjoy).
Now, if you look at things from the outside, a sports league is competing with all other forms of entertainment...capital flows to whatever entertainment product is most compelling. From this perspective, the entertainment market is capitalistic.
The socialized league revenue and labor price fixing (salary floor/ceiling) makes the teams compete on strategy, smarts, work-ethic, etc...this, IMO, yields a much more compelling product overall.
All that being said, it's important not to conflate the benefit of the "socialized" sports league model to what applying the same model at a region/nation does.
What is "problematic" about pro sports in the US?