As long as I jumped back into the mudbath, you wrote: “…he can ban anything he wants on his property, including his parking lot. Forcing him to do otherwise is a violation of his rights.”
It looks like you’re saying that if he wants he can ban the life and liberty of others; that if they are on his property, the owner can kill someone at will or prevent someone from leaving. If you’re not saying that, then he can’t just ban “anything he wants” on his property. That raises the two part question: Can he ban anything at all and if so what circumstances justify doing so?
We all agree the answer to the first part is “Yes.”
Agreement by all on an answer to the second part seems to be a problem.
I can see why it might look that way, but that isn't how it actually works out. As the owner he can refuse you entry; as long as you aren't on his land, he can't touch you. If you attempt to force entry, he can indeed defend himself.
Supposing you are on his property because you were allowed on, he can't defend his property against your "trespassing", because you aren't trespassing. What you're suggesting is that the land owner might now say, "I want you off my land in one second... ONE! OK, now you're a trespasser!" and start shooting. He cannot do that because in allowing you onto his land, he made an implied contract that he would not molest you, including assault or imprisonment. He volunarily waived his right to call you a trespasser when he invited you onto his land.
He can of course order you off his land, but the contract implicit in his originally allowing you in, means that he must give you sufficient time to get off his land. If you're standing right at the property line, he might legitimately give you the count of three before he starts shooting, but if you're deep in his land somewhere, he must allow you to depart.
...then he can’t just ban “anything he wants” on his property.
He can, but only with advance notice. For example, he might invite you onto his land on the condition that you sign a waiver, acknowledging that the landowner will shoot you if he feels like it. You'd be a fool to sign the contract, but if you signed it and entered his land, he'd then be within his rights to shoot you. If, on the other hand, he invites you to dinner and up and kills you, he's a murderer: his invitation implied a contract not to harm you during your visit. That isn't a limitation on his property rights; it's a case of a voluntary contract waiving certain rights.
Can he ban anything at all and if so what circumstances justify doing so?
He can make absolutely any rules he wants, but he can't then change the rules at a whim. This answers your original objection, but leaves open the possibility that he might lawfully organize a "paintball" game using live ammo.
The law doesn't work that way today, however. Courts do not respect the liberty of contract, so for example a contract to provide assisted suicide, or sexual services, would be nullified by the courts. But I think it was already clear to you that we're discussing "shoulds", not current US law.