I think that when an apology is not promptly offered, especially for an egregious mistake, it's very telling. People know when they fuck up or do something wrong. (Certainly in more extreme situations at least.) And when it's not offered, the relationship should suffer because clearly the one person isn't willing to acknowledge they did something wrong for whatever reason - because they're a coward or because they're too proud. (Ironic how it's both sides of the spectrum.)

If I don't agree with someone who is raising something I've done to make them upset I will tell them I am sorry they feel that way.

I do apologize. I just only apologize for what I feel is merited. At that point I have acknowledged the other person's ruffled feathers and it's about "where do we go from here," if it's a person I even want to have that conversation with.

There are truth be told a few people (2) who I would not even say "I am sorry you feel that way" to. But they are people who I no longer care about how my actions make them feel