In the United States, many people find it cringy and even offensive when someone says "I'm sorry," especially when he or she does it frequently. Did the Buddha or any Buddhist teachers have any guidance on this point?
The purpose of saying "I'm sorry" has somewhat gone astray nowadays. Too often it has become a token utterance bereft of heart. The Buddha points out the key aspects of dealing with transgressions:
The Buddha also teaches beyond mere apology, and AN5.142 is well worth readying and studying in its entirety. Especially take note of the teaching on regret:
AN5.142:6.4: ‘Venerable, the defilements born of transgression are found in you, and the defilements born of regret grow. You would do well to give up the defilements born of transgression and get rid of the defilements born of regret, and then develop the mind and wisdom.
It's praisworthy if one sees a fault as fault, and confesses it, ask for pardon as well. Not seeing a fault as fault, not confessing it, not asking for pardon, such one is called a fool, like someone who rejects a pardon, good householder. Reconciliation, Right & Wrong gives further food for release on this topic.
"As for the times when you realize that you’ve harmed others, the Buddha recommends that you understand that remorse is not going to undo the harm, so if an apology is appropriate, you apologize. In any case, you resolve not to repeat the harmful action again. Then you spread thoughts of goodwill [mettā] in all directions.
This accomplishes several things. It reminds you of your own goodness, so that you don’t — in defense of your self-image — revert to the sort of denial that refuses to admit that any harm was done. It strengthens your determination to stick with your resolve not to do harm. And it forces you to examine your actions to see their actual effect: If any of your other habits are harmful, you want to abandon them before they cause further harm. In other words, you don’t want your goodwill to be just an ungrounded, floating idea. You want to apply it scrupulously to the nitty-gritty of all your interactions with others. That way your goodwill becomes honest. And it actually does have an impact, which is why we develop this attitude to begin with: to make sure that it truly animates our thoughts, words, and deeds in a way that leads to a happiness harmless for all."
~ Thanissaro Bhikkhu "Metta Means Goodwill" https://www.dhammatalks.org/books/BeyondAllDirections/Section0007.html
The blamer, afflicted with this view (and suffering from it), but still detached enough from the view to question it, would do well to acknowledge and reflect on their uncertainty about it. Because that is what will allow them to take a step back and see the view as unskillful, stressful, harmful and unworthy of sustaining. When they see the view clearly as the origin of the problem (the stress and suffering that they are feeling), then they can drop it. It will still exist, but they no longer identify with it as "me" or "mine".