I’ve always thought of it as being the opposite. I only ever use ‘I apologize’ or ‘my apologies’ when I’m sorry about something I did, when it’s my fault. I use them because it feels more sincere to me, and more effectively communicates my intention.
Eh, it might be a regional thing, but I agree with OP. 'I apologize' feels kinda snarky. But then again, as a Canadian, saying sorry is kinda one of our national pastimes.
Yeah, that's my take. "I'm sorry" is about how you feel. It's like the OP post: you're expressing sympathy, or remorse - an emotion. If it's because of something you did, it does double duty as an apology. "I apologize" gives no indication that you're sorry about what you did, or concede that it was wrong. It's merely a recognition that harm was done, and giving a concession to the other person to achieve harmony.