I don't think it necessarily has to do with how they have some sense of self-loathing, but it just goes back to a traditional sense of gender roles. Of course, these traditional gender roles are exemplified by The Bible where the push is for the man to be the head of the household and the woman to support her husband. That's likely where this woman's idea of household voting comes from; essentially, if the man makes the decisions in the house, why should she vote (i.e. make a decision)?
To be clear, I think it's absolutely ludicrous, and one thing that I abhor is the obsession people have with "tradition". Just think about how often we do things "because it's tradition" without realizing why we actually do it. Buying a diamond ring for an engagement? Tradition! ...or the result of a 1930's marketing campaign from De Beers to push the idea of a diamond representing love. De Beers thanks you for your patronage.
Although, I think I hate traditions like the diamond one mentioned before the most. I can understand that some traditions actually
had a purpose at one point in time (and may just be archaic now), but marketing leading to tradition feels like dirty manipulation. In other words, there never was a purpose other than lining someone's pocketbook.