From the time they get to the stadium to the time they leave, it's all part of playing the game.
No, they are only playing the game when they are on the field. Anything else, that's at their discretion.
Now, I'm sorry you don't feel comfortable that they are protesting, but that's your problem.
They don't just get paid for game time. If a player never plays in a game, he still gets paid. NFL players have to attend practice, team meetings, workout sessions, ect. When they are in uniform and on the field on game-day, they are representing their team and employer.
I think it is ultimately up to the team owner, if he wants to allow it, then I think it is OK. He will lose a lot of revenue because of it, but it is within his rights to allow it.
Otherwise, employees typically don't have a "right" to engage in protests while "on the job". Do you see Best Buy employees protesting while at work? Do you see police officers or government employees protesting while at work? No.