You are right in what you say about things to consider when making a decision, because some birds certainly don't usually need our help.
I once received a visit from an injured pigeon, it had a broken wing, not the bone, but the flesh, because it was bleeding.
It was at home for a couple of weeks and I fed it, even ate with my cat, but one day I woke up and didn't see it, so I guess it flew away.