I believe Ms. Doyle is burning a straw feminist. The real issue is essentialism. Trans people are essentialists. (Some protest that they aren't, but their arguments seem to me word play.)Trans people are essentialists because they feel that something is essentially amiss with their birth assignment. What are they feeling? (The question is rhetorical; I assume the answer defies words, or at least prose.) Whatever they feel, it must be a feature of their self-identified sex, which means that there may be some sense to a social system in which most happily cis members of each sex adopt certain roles suited to their sex's essential nature. One can see how that would upset a certain school of Radical Feminists.

TERFism has nothing to do with defending the patriarchy. On the contrary, TERFs argue that trans people validate the patriarchy by making it at least possible that male domination is a natural outgrowth of the gender differences dictated by the essences of the different sexes. For those into genetic fallacy (pun very much intended this time), I will disclose that I'm a "Vive la difference!" cis male. But I do try to call 'em as I see 'em.