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.