All people living in the USA are 'limited' by our supreme "Law of the Land".
Thus, - the Constitution puts limits on people, as well as on governments.
You really think that "the private sector" can ignore our Constitutional freedoms, our Law of the Land?
The marketplace itself is restricted by our Law of the Land, by our freedoms as per the Bill of Rights.
Any member is free --- to start their own jockey club where you can name horses Mohammed, Roosevelt or whatever suits your fancy.
Do you really think a State or local government could or would [constitutionally speaking] license a racetrack that allowed gambling on horses named Mohammed, Roosevelt or whatever offensive name suits your fancy?
Dream on.
They would deny the license on some other constitutionally solid ground, but they would never admit to denying the license over a "free speech" issue.......