I think maybe you’re not picturing how such a regulation would work in practice. Suppose someone who looks like a woman goes into the women’s bathroom. Do you ask them if they have a penis, or do you just let them use the bathroom? To enforce the regulation that people have to use the bathroom corresponding to their biological sex, you would have to continually ask people about their genitalia. In other words, you’d have to continually take an interest in whether any given woman is a transwoman or a cis woman.

Also, I didn’t quite say that increasingly precise distinctions of sexual and general identity are a response to conservatives. I said that it’s because of conservatives that these distinctions are thought to be (and in practice are) very important. If we lived in some kind of free-wheeling hippy society, then these aspects of people would be of the same level of interest as, say, their dietary preferences.

As things currently stand, if you are a woman with a penis, there are lots of people ready and waiting to tell you that you're not really a woman. It's hardly surprising that transwomen take their gender identity very seriously when it's constantly under attack. Nor is it surprising that they construct new categories (e.g. 'transwoman') when the existing categories are so rigidly policed.