On a hunch I typed "Ross Douthat" "Phyllis Schlafly" into Google to see if he described Good Ol' Phyllis as a feminist. Yep. In that very article. Phyllis Schlafly described herself as an anti-feminist.
The logic contortions that go into stating, without irony, that "feminists" like Barrett recognize that they wouldn't be able to hold the jobs they hold without the women's movement, but they nonetheless reject all of its central tenets, would make the cast of Cirque du Soleil wince.
I can walk you through it. "Fairness" is a big conservative principle. If anything, I would say that conservatives lean into fairness as hard as liberals lean into empathy. So whatever bleeding hard liberal bullshit free government healthcare saves Mitch McConnell, it's only "fair" that they use it since it's available to everyone. If they take it away, that too is "fair" because there's nothing preventing anyone from using the same perks and advantages they have. And in general, Republicans got to where they are by working hard, so it's only fair that they reap the benefits. If someone else worked harder but didn't make it quite as far, well, that's as fair as it can be because as we all know, life isn't fair so we better make sure we all get our fair shake - I mean, the Emancipation Proclamation made it so that all men are created equal and if white men get further than white women, it's because on a level playing field they just do better. I mean, you wouldn't really have Amy Coney Barrett turn down the advantages that are availble to people like Alexandra Ocasio Cortez, would you? And look - she made it all the way to the House of Representatives! Clearly the system is working for everyone. The fact that Herman Cain was a successful black conservative entrepreneur is proof positive that anyone black can be a successful conservative entrepreneur, the fact that he's the only one has a lot more to do with how traditionally lazy black men are than any systemic problems of equity. It's that easy.