Pro athletes do have contracts that may limit their free speech rights or other activities while they are under contract since they are representing the team and are public figures.
Of course that applies to athletes on any side of a public issue.
If the team “picks sides” and goes after folks expressing themselves on one side of an issue and not another then a good lawyer could tear them to shreds.
I guess have mixed feelings on the Clevenger/Kaepernick deal. I wanted Kaepernick to be told by his team or the NFL pretty much what you said - that the team and the league do not want to be part of his or anyone else's private causes. It seems unfair that the Mariners lowered the boom on Clevenger in light of Kaepernick getting away with it. But, they saw how the Kaepernick thing has snowballed in the NFL. If teams are perceived as coming down on one side of an issue they are going to piss off some portion of their fan base. For example, it is hard impossible for me to root for a team when I watch the players disrespect the flag.
This kerfuffle sort of reminds me of another forum I hang out at called the Home Theater Forum. I have no idea if the people who run it are conservative or liberal. But what I do like is that they don't allow political comments at all. They only want the site to be about home theater. They have off-topic forums but they cannot be political. If someone puts in a political jab (usually a snarky liberal), they delete the comment and issue a warning and sometimes ban the poster. I really appreciate that approach. I had to leave the Sinatra Family Forum because it was so full of DU types from Nancy S on down through most of their membership.