So you agree he has a "blanket protection" for free speech in his private name for his horse?
It is really extra-Contstitutional as to whether he wants to call his horse this or that.
You just agreed that to race at a public track, the jockey club could [constitutionally speaking] limit what he 'calls' his horse. - You can't have it both ways.
The Constitution is meant to put limits on government, not people........
All people living in the USA are 'limited' by our supreme "Law of the Land".
As servicemen, we swore an oath to support & defend. ALL of us are so obligated, as per our oath of Citizenship. -- So -- We the People can't run about calling publicly raced horses by names that may be offensive to religious, political or ethnic groups. - Those are 'fighting words' under common law.
It's a private track, open to the public. There's a difference. Here's a personal example: I work for a private company (no publicly traded stock). For a few years after 9/11 I had a sign, that I made myself, on my work bench that said "I SLAM ISLAM". Like I said, it had been here for at least 3 years. One day the boss/owner of the company asked me to take it down. I did, without protest. His reason was that he was afraid someone might see it and be offended. This is a small company, 15 employees and most of them are related to the family that owns this company. We are all Protestant and Catholic. We get few visitors as we are a small business electronics company that builds primarily stuff under government contracts, US and foreign. We hardly ever even see a "customer". Just build, test, ship the product and get the check. Now I could have gone on about my "free speech rights" etc, but this is not my company, and he has every right to tell me to take down my "political sign", because it's HIS PRIVATE PROPERTY, even though he agreed with my message!