"I'm sorry you're upset"
There's a sorry for other people's feelings. A friend is having a bad day and you say sorry. A colleague is stressed about a deadline you had nothing to do with and you say sorry. Your partner is cross about something at work and you say sorry, and they say "it's not your fault", and you say sorry for that.
Look at what the sentence does. "I'm sorry you're upset" takes a feeling that arose in someone else, for reasons that are theirs, and places it in the frame of your responsibility. You've said sorry, so there must be something to be sorry for, so their upset must in some way trace back to you. You didn't mean that. But the word means it, and the listener hears it, and one of two things happens.
Either they correct you. "It's not your fault." And now they're managing your apology in the middle of their bad day, which is one more thing on it, and they've learned that telling you about a bad day produces an apology to be dealt with.
Or they don't correct you, and over time they come to feel, in a way neither of you would put into words, that their moods are somehow your department. That when they're upset, you'll say sorry, and the saying sorry means you've taken it on, and they can lean. That's not kindness. It's a slow transfer of responsibility for another adult's feelings onto you, one sorry at a time, and it's heavy, and it was never yours to carry.
There's a sentence that does what you meant. "That sounds rough." "What a day." "Do you want to tell me about it?" Those say I've noticed and I mind, which is what the weather sorry was trying to say, and none of them takes the blame. Brooks's finding in chapter 5 was that the trust came from empathic concern, and empathic concern doesn't need the word sorry to travel. It needs you to have noticed and to say so, and "that sounds rough" is noticing said aloud.
So the sorry for their feelings can go, and the noticing can stay, and the friend with the bad day gets a person who's listening instead of a person who's apologising for the weather in someone else's head.