The counter-claim is not that racism is exclusively a men’s issue. The counter-claim is that the claim “men’s rights don’t vary by state” is false, as evidenced an example of how men’s rights do vary by state. The implied part that should have been explicit is that the way racism manifests from state to state also has gendered aspects, with some disproportionately affecting women (e.g. hair/dress policing in the workplace) but some also disproportionately affecting men (e.g. incarceration). That is to say, racism and sexism are intersectional. Another example might be how custody rights typically vary from state to state often unjustly disfavoring the father, given all other things being equal.

I’d suggest that this argument does not go against the underlying position of OP that “patriarchy bad”, rather it corrects OP to highlight how institutional sexism typically falls along normative/conservative conceptions of gender for men too. That is to say “patriarchy bad mostly for women, but also bad for men too”.