> so allowing trans men into women's bathrooms basically ensures that all men can freely enter them, claiming to be trans men.

Any examples of this hypothetical situation actually happening?

If these spaces were regulated by law, and there was a realistic prospect of penalties for males who choose to impose themselves on female spaces, then this would have a significant deterrent effect. The males who say they're women (including males with a transwomen identity and your hypothetical set of males who claim to be transmen) would refrain from intruding.