If we are going to insist, as we should, that nobody is above the law, then this is indeed appropriate IMO.
She certainly could have asked to be given just a warning, verbal or written, but to try and use her position is just obnoxious!
It's very common in Florida for people to use golf carts as personal transportation. This will be an unpopular opinion, not that I care, but this is a very minor offense. For her to lose her job because she asks for professional courtesy is unfair imo. Nothingburger.