I think some of the problem is that we've thought about this a lot, our arguments have refined and developed (I cringe at my 20 year old self, who would have breezily said that I had no problem with prostitution or surrogacy), whereas, for a lot of men, this is the first time they're hearing this stuff, and on the surface it can seem to make sense.
My DP listens to Joe Rogan, which is generally 2-3 men sitting waffling on about stuff they don't necessarily know a lot about (not always, he has experts on subjects too sometimes). More than once I've overhead what's being spoken about, and stormed into the livingroom to suggest that he stops listening to this rubbish, and told him exactly why Joe et al are talking bollocks.
We've had late night conversations on pornography, which over the course of a few weeks really made him think (reading out loud the titles of videos on the front page of a major aggregator really hit home)
I would like to think that our talks would mean that he'd see through Peterson's shallow analysis, but who knows - it doesn't directly affect him, so he wouldn't think deeply.
When it comes to your DP OP, you're going to have to decide how you want to handle it. I know that if mine wasn't willing to discuss, if he started spouting MRA shit in front of the kids, then yes, actually, I really would be looking at leaving - I'm not going to live with and support someone who thinks that way.
And yes, I do like the suggestion of swapping roles. It exposed a huge flaw in our relationship when I wanted to go back full time from freelancing, and opened the discussion on whether he would be doing drop offs or pickups to enable me to work (he had flexitime, this was completely within his ability to commit to), and he flat out refused to take any responsibility for them. It nearly broke us.