It would be У этого здания, but I think В этом здании is preferred since it is specifying the elevators are actually inside the building. Theoretically у этого здания could imply that the elevators are against the building but not inside, so it's slightly more ambiguous. Typically places/non-animate objects that have things tend to have them в... себе as opposed to у себя, from what I've seen.
I completely agree with you about this paritcular sentence, but I disagree with your animate/non-animate statement. E.g. "У этой машины четыре двери" sounds better to me than "В этой машине четыре двери". I think the actual difference is whether Y is an integral part of X or whether it's some internal feature. E.g. "У этого здания четыре стены" (this building has four walls) but "В этом здании четыре лестницы" (this building has four staircases).
There is an exercise "В лифте зелёные стены" (https://forum.duolingo.com/comment/15464948), where В is used.
What's the difference between the walls of a building and the walls of an elevator?
I think I'm still confused with the difference between "an integral part" and "some internal feature".