Not surprised in the least about Element unfortunately, it really screamed for a rewrite (one of the biggest problems was the horrible choice of database, which unfortunately was very hard to replace)
Sounds like the old Element app that was being used was using an always on notification to poll for push notifications instead of using a push service.
Element X only supports FCM cloud push notifications right now I think
Correct, the novel thing Element X does different is sliding sync which allows for much faster partial syncs when the app is opened. The classic apps would have to do a much more substantial (and energy intensive) sync each time it was brought up to foreground.
The big drawback are the X versions are still very far from feature complete.
Element X only supports FCM cloud push notifications right now I think