Trans people already have equal rights. The contentious question is on whether they should have additional privileges, e.g. a subset of males being permitted to use female-only spaces on the condition that they say they are women.

They won't have equal rights if they continue down the path towards criminalizing the act of a trans person reading a story to a child (example from TFA).

I think that is not correct, the example cited in the article is about drag queens performing a 'story hour', not trans people.

I give zero percent odds that the person making this complaint was aware of the distinction between a cisgendered male in a dress and a trans woman in a dress, but you're right, the demographic under threat is more broad than I let on.