I was feeling pretty frustrated and hopeless on Tuesday, but since then, progress has been made.
➕ A friend of Uncle agreed to take his accumulated mail to a shipping place and ship it to me to deal with.
➕ The ticket I filed with the postal service elicited a phone call from a supervisor who said there was a "replacement mail carrier" who did not know there was a forwarding order for Uncle. This is being fixed.
➖ Unaddressed: is this going to recur the next time the letter carrier changes?

Worth mentioning: the other reason why it's so important for Uncle's mail to be forwarded to me rather than going into his mailbox, is because his usual m.o. is to occasionally check for mail in his mailbox, bring any he finds up to his apartment, and then ignore it.
Before I set up mail forwarding I had to periodically travel from Boston to NYC just to go through his apartment and search for piles of mail that needed to be dealt with that he'd done nothing about.

@jik Are you using premium forwarding, or regular? Premium forwarding isn't cheap, but when I was doing the snowbird thing it was MUCH more reliable and faster than piece-by-piece forwarding, and 100% worth the price.

@callisto I am not using premium forwarding, for two reasons:
1) The forwarding I have set up right now is temporary rather than permanent, because I'm worried that if I set up permanent forwarding the entities that deal with Uncle will update his home address to match mine, which could wreak havoc on his government benefits. I can't risk that.
2) I don't want packages to be forwarded. I need people to be able to ship stuff to him.