Movies you buy/rent in Google Play Movies are actually available in YouTube as well. There's a section in YouTube called "Purchases." I've used this to stream Google Play Movies purchases on my Fire TV. I would imagine the same would work for an Apple TV since it has the YouTube app.
I know it is a cumbersome workaround, and that a company as big as Google really should have a product that just works, but since you mentioned that YouTube works fine, it is a possible workaround that might save you 35 minutes next time.