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!
All people living in the USA are 'limited' by our supreme "Law of the Land".
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.
It's a private track, open to the public. There's a difference.
Fine. - We agree then, that [constitutionally speaking] at a private track, open to the public, racehorses names can be limited.
Thus, - the Constitution puts limits on people, as well as on governments.