But how to say in a polite and effective way, "you're behaving irrationally, the offence is something you created for yourself; I feel bad that you're feeling bad now, but I did nothing I should be apologizing for"?
I sometimes find myself tempted to use a non-apology like "I'm sorry if I offended you" in order to short-circuit the irrational anger in the other person quickly, so that we can get to the point of actually figuring out what I've said that triggered the problem and how their reaction was wrong, and what's the deeper issue underneath.
"I'm sorry I offended you" is a perfectly reasonable way to express "I feel bad that you're feeling bad now."
I personally don't understand why you'd feel bad about how another person feels as a result of something you've said, but not feel that you should apologize. To me, the two go hand in hand. It's not about "should," it's not about right and wrong, it's simply about expressing the sentiment that you regret what happened as a result of your actions.
Even if you said something totally innocent like "I like apples" and somebody got offended at that because they're crazy, you can still sincerely apologize for that if you actually care that they feel bad.
And of course you don't have to care that they feel bad. In many cases not caring would be a perfectly reasonable response. In which case, don't lie by saying sorry.
Well, I think a lot of English speakers are confused as well. There seems to be this idea that "I'm sorry" implicitly means, "I fucked up, it's all my fault, I'm a worm, I deserve to die," and so forth. But it's really just, "I wish it wasn't so, and I'd do it differently if I could go back and try again," or something along those lines.
Except, under those circumstances, I'm probably not losing any sleep over the fact that I offended you. So that statement means I'm shouldering the blame at some level but it's also typically dishonest. I'm not actually sorry I offended you because I think you're being an oversensitive idiot.
Then don't apologize. If you're not actually sorry, don't say you are.
Edit: it occurs to me that this hooks straight back in to the article's question. Why are most public apologies so bad? Because most of them are made by people who aren't actually sorry.
I sometimes find myself tempted to use a non-apology like "I'm sorry if I offended you" in order to short-circuit the irrational anger in the other person quickly, so that we can get to the point of actually figuring out what I've said that triggered the problem and how their reaction was wrong, and what's the deeper issue underneath.