The First Amendment is a limitation only on government power. It prevents the government from restricting speech. It has no impact at all on private enterprise. So it's not that a private enterprise can suspend First Amendment rights. In private enterprise, there are no First Amendment rights to suspend. It's a subtle but important distinction.
They can certainly fire you for disruptive behavior but you still have a right to speak if you choose to risk your job. And I support that view as the rest of my post makes clear.
Yes, I wasn't disputing the conclusion of your post, only pointing out that there is no right to free speech in private enterprise. The football players do not have any Constitutional right to do what they're doing since their teams and the NFL are private enterprises. Unless there is a clause in their contracts giving them them the right to do it, they can be fired without repercussion.
I should have been more clear asserting player rights to free speech since this has been the battle cry of the left in their support of CK. LBJ, Curry, CK and the rest can tweet all they want on their time but leave it there when you come to work just like ordinary Americans must do.