If he is the agent of the property owner, yes, he is legally in the right. There is no law that says the rules can't change.
Exactly.
The sign sets one rule. A representative of the property owner sets a broader rule.
There is no issue, pardon, there should be no issue. It's private property. Their rules, written or verbal. Don't like their stance? Find a hospital more to your liking.
Personally, it seems like a silly thing to start an argument about.