"Responsibilities" in his take is code for "being forced by the state to do stuff." I don't think rights should necessarily come with responsibilities, let alone a responsibility to go kill people (and get death, dismemberment, and/or PTSD), unless having that right logically implies some kind of responsibility (for example, the right to universal healthcare requires everyone pay taxes).

And sure, you could draw some logical link between the draft and the right to vote (e.g. "you're not guaranteed any rights if some other country takes over yours"), but still, women should have the right to vote because they, too, live under the system that voting influences. Even when some right logically corresponds to some responsibility, it's not necessary to give only those people who have that responsibility that right--there's no reason it has to be a one-to-one coupling--and also, women have responsibilities within society to obey the law, which is influenced by voting, whether they're subject to the draft or not.

This hot take just sounds like a sophistic way of arguing for the oppression of women.