"It'd be humiliating"
Strip the reasons away and this is often what's left. The climb-down feels humiliating. Not looks. Feels. There's heat in the face and a drop in the stomach and a sense of being visibly smaller, and the sentence "you're right" has to be pushed out through all of it. Holding avoids the feeling, and avoiding a feeling that strong doesn't need a reason.
Karina Schumann, reviewing the research on why people don't apologise, names three barriers, and they map onto the climb-down closely. The first is not caring enough about the other person or the relationship. The second is the threat to how you see yourself. The third is believing the apology won't work anyway. Most of the beliefs in this book so far live in the third one. This chapter is about the second, and it's the one that isn't a belief at all. It's a feeling, and arguments don't reach it directly.
What reaches it is looking at it. So look. How long does the feeling last? Not the imagined version, the actual one, on the occasions you've climbed down. My observation, and I'm offering it as that, is about a minute. There's the heat while you say the words, a beat of silence after, and then the other person says "no worries" or "I'd forgotten too" or changes the subject, and the heat goes. The minute is unpleasant. It is a minute.
Now look at the alternative. Holding doesn't avoid the feeling. It spreads it. There's a low version of the same heat under every defence, because you know the defence is false while you're making it, and that version doesn't end when the argument does. It's there in the silent mile, and in the car home, and at two in the morning when the roadworks come back to you. You've traded one minute of the full feeling for an evening of the diluted one, and the evening is longer.
The feeling is real and it's not going to stop arriving. What changes with practice is not its size but its meaning. The first few times, it feels like being lowered. After a while it feels like the moment before the argument ends, which is a thing you start to want.