I think they have push notifications in XMPP these days. At least Prosody has modules like mod_unified_push and mod_cloud_notify and that seems to be supported for example by Conversations.im

To be honest, I didn't have lots of battery drain, back when I used XMPP. And other old-school protocols like e-mail and sip voip don't seem to be very bad either with whatever mechanisms they use. Or my phone isn't reporting battery drain correctly.... And with Matrix I also had to set up push notifications manually, or it'd just receive messages with a random delay per default.

Not at all, message archive management and Push Notifications are some of the core XMPP extensions that almost any XMPP app supports.

Let me tell you an other huge advantage of XMPP for those who care about privacy: it's called Omemo, which is essentially the same protocol used by signal. Given the current push by govs to undermine encryption and private messaging providers, it is probably a good idea to look into self hosting anything remotely related to privacy. Even if they manage to (and they will) to pass their spyware everywhere, if more people self-host it would be much more costly for them and we would see many more secure protocols popup.