"Everyone will remember it"

The audience is the reason holding feels like protection. If nobody were watching, you'd say "oh, the roadworks, you're right" and take the third exit. It's the eyes on you that make the turn into a threat, and the belief is that those eyes are keeping a record.

Look at who's actually there. In the car, it's one person, who has already forgotten which exit you argued for by the time the kettle's on at your in-laws, because they're thinking about their mother. In the meeting, it's six people, four of whom are looking at their laptops and one of whom is wondering whether the meeting will overrun. In the group chat, it's nine people, seven of whom haven't opened it. The audience you're holding for is the one in your head, and the one in your head is paying much closer attention than any real one.

There's research on why your own climb-down looks worse to you than the same climb-down looks to anyone watching. Anna Bruk, Sabine Scholl and Herbert Bless ran seven studies in 2018 on what they called the beautiful mess effect. People imagined themselves, or others, in situations where they'd have to show something they'd rather hide, admitting a mistake among them, and rated how it would come across. People judged their own showing much more harshly than they judged others doing the same thing. The explanation the researchers give is that you see your own admission up close, all the wobble and the heat in your face, and you see someone else's from a distance, where it just looks like honesty.

Test this on yourself the other way round. Think of the last time someone climbed down in front of you. A colleague who said "actually, I've looked again and you're right about the dates." A friend who said "no, I had that wrong." What did you think of them? My guess is that you thought slightly more of them, and that you'd forgotten it within a day, and that if you're struggling to think of an example at all, that's because climb-downs don't register as events. They register as the argument ending.

The audience will remember the argument if it went on for twenty minutes and ended with you both silent in the roadworks. It won't remember a climb-down, because there was nothing to remember. That's what you're protecting yourself against.