Another thing they might be thinking is they think you want them to leave (owing to the number of disagreements and the amount of criticism which they interpret as a kind of rejection) and they find it hard to understand why you aren’t agreeing with them leaving. Maybe it’s a kind of tribalist thinking which they project onto you - they think they’ve been rejected from the “in” group and want you to at least agree with that.
I guess they could end up thinking they’re stuck/trapped, and don’t know what path will lead to some sort of agreement that they’re willing to pay for (i.e. that they can reach in an amount of time they want to spend) and having some sort of agreement would help close the thread in their minds.
I guess there’s part of me that’s worried about this possibility here - that I want to leave a conversation because I disagree but don’t know how to proceed with that.
I checked through your intuition-related articles and this one seems to be very relevant.
I think when people have arguments sometimes they think (or at least suspect) the other guy is being a jerk. In that sort of situation it can be a bit scary to say something like “I have an intuitive conflict”. It’s kind of like a situation where someone is scared of finding out their test results for a scary disease, even though it’s a lot better to know if it’s there so treatment can start. If someone suspects an interlocutor is a jerk, saying “I have an intuitive conflict” is a jerk-test, and they might be a nasty jerk in response. Even though that would then allow the treatment to begin (i.e. ending the conversation and moving on) it’s still kind of scary to do.
I don’t think I fully understand that kind of fear. I have things like that which I sometimes avoid before doing them later (either because it stops being scary, or because I’m under time pressure and make myself do it).