It's not wholly a smear, though some certainly is. Let's take this one by one.
Kirk explicitly believed wives should submit to their husbands. He stated this multiple times, including in some marriage public discussions w Erika. She likewise believes this and has preached it on her podcast Midweek Rise Up.
I have read a lot of various kinds of conservative Christian thought, and it is unusual for all but ultraconservative/explicitly misogynist types like 'redpill Christians' to say that women should not be allowed to lead AT ALL. Most believe that women are only barred from leadership in the church and in marriage. I strongly disagree with this, but I also dislike people exaggerating what people believe. It makes it harder to spot actual Christian extreme misogyny, which does exist.
That said, there was certainly opposition to Sarah Palin's run on account of gender by people who might have fewer issues with someone like Bondi or Coney Barrett who does not hold ultimate power.
It's interesting to note Erika herself has a job preaching (I need to check). So women speaking about religion is OK, but not priesthood or church leadership roles.
As to Kirk's view on women and careers, I don't think he was explicitly opposed to them. He emphasised the importance of having marriage and kids first, which has pros and cons imo, but he didn't say it was forbidden.
That would be hypocritical anyway, when his wife has the podcast, clothing line, charity work, modelling & acting work & preaching/speaking gigs.