It's true that it's a big hunk of JSON. And I'll readily concede that when efficiency matters, I'm more partial to binary formats like CBOR. But XMPP is XML, which... isn't exactly lighter.
There's another HN thread where I've talked more about XMPP vs Matrix here: https://news.ycombinator.com/item?id=9772968 -- long story short, I tried to write an XMPP client, and I got grey hairs, fast. The story for consistent delivery is pretty much bananas. (And huh, based on that date, I guess I've been a Matrix convert for almost TWO years now! Time flies.)
I'm not entirely sure what the comment about battery is about either. If anyone has a networking technology that can work without turning on the phone radio, I'm sure we'd all love to hear it...?
Practically speaking, I leave the Matrix Vector app open on my phone constantly. As of this morning, I have half a charge, and three more days of charge to go. I don't have any other apps to provide a good comparison here, but I'll leave this screenshot here for what it's worth: https://matrix.org/_matrix/media/v1/download/matrix.org/IgkU... Looks like my phone has spent about the same order of magnitude of battery on just plain paging the cell towers as I moved around the city. As other comments on this thread cover, if you use proprietary Google stuff, you can wake the phone up slightly more efficiently. But whatever this app is doing works pretty fine for me.