That's BS. Apologies are for when we hurt someone or do something wrong.
(This is at the idea, not trying to attack you)

If I thought my T said or did something hurtful because he didn't care or wasn't trying then I wouldn't be seeing him in the first place. I should be able to assume he's doing his best (and I feel the need to clarify that "best" =/= most effort, boundaries are involved too, it includes taking care of himself).
Apologies are for when he does his best and I still get hurt.
He apologizes for times he's done nothing wrong but I was hurt by something, like there's no way he could have known that something would upset me and I'm acknowledging that I'm being irrational and unreasonable, and that kind of surprised me and I don't think it's necessary.
And he apologizes for little mistakes as well as a really big one he's apologized for many times.

I think with the poem I might be annoyed, but whether or not I saw it as a red flag would depend on the bigger picture. It would be one of many small things that would affect my opinion.