I think you could change "a woman" to "[someone]" using those editorial bracket things and the pronouns won't be weird.

That's probably what I've done, too.

(I also feel like you get a similar effect if you change "woman" to "lady" and I have no idea why.)

(I'm not a native speaker, so don't trust me about this.) Using “woman” suggests that the only salient feature about that person was her gender, which is indeed kind-of weird IMO; OTOH, using “lady” (or “girl”) would suggest that her adult (or young) age was also salient, and that would lower my estimate for how strongly the out-group homogeneity effect affects Rod Dreher when he thinks about women. (Also, I'm under the impression that many of the stereotypes about women are closer to the truth in the case of younger women than in the case of older ones (as an ageing effect, not a cohort effect), though this might be due to selection effects in the groups of people I interact with.)