I think this way will get him to eventually stay in his bed though, as he's not coming out and just being put back, yk. There's no incentive and he's having to do the work himself, so it's more of a learning process, hence the screaming.
Does he have a cup of water in his room that you can offer instead of milk? I think it's fine that he falls asleep uncovered. You can dress him warmer if needed, or just go back and cover him like you did after he falls asleep. And of course, like Anna did, eventually teach him to cover himself.
(((Hugs))) MM, I think you have a good plan.