The best insight here is that self-worth and fear of shaming are the main motivators for not agreeing. A weaker argument, that is less threatening to self-worth is one way of preventing that your discussion partner goes on the defense. Another way is to make him feel safer in the sense that he can trust that you will not hurt his feelings. This can be done by affirming that you are on the same page with him, that you…
Could it be that the tendency to preserve the coherence of one's views goes beyond self-worth? E.g., most people, I think, would try to avoid holding in their minds ideas which don't agree with each other. So, if your previously acquired ideas are already leaning this way , they may tend to oppose new ideas that are leaning that way .
As I read about it, I could remember several instances in my life where I have behaved exactly as described.