My brother apparently thinks there is no reason to apologize for anything other than an artificially imposed sense of guilt, shame, and defeat, along with submission. While this is usually the only reason most children apologize, I do not believe these are valid reasons to apologize at all. It shouldn’t be just out of a sense of coercion or submission.
It should be out of a sense of genuine remorse for one’s actions, a real desire to admit wrongdoing and to make amends. Or, to paraphrase Anton LaVey, when you’ve painted yourself into a corner and the only way out is to say “I’m sorry”, “I’ve made a mistake”, or “I wish we could compromise somehow”.