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If trans-men wouldn't endanger themselves by doing this, I would encourage them to chill in women toilets.
But they shouldn't put themselves in harm's way just to prove a point.
We should just treat them right and protect them from this nonsense.
We should be prepared for a fight on that front. The current uproar is about trans women. Put a trans man in power (anywhere) and the shit is really going to hit the fan. The rhetoric will shift from pseudo-sexualized bathroom nonsense to "how dare you disrupt our patriarchy."
Transphobia both currently and for basically the last 50 years at least has always been framed and rooted in the idea of protecting women from predatory "men". Trans men have been a footnote, at most and I doubt one becoming prominent will have the impact you expect.