the push notifications are a centralized service that does go through their servers, even when you self-host the chat itself.

Yeah, but everything else being equal (using Element), I have my account on their server too, and use push messages there. Or I only use their push messages and host my own server. As far as I understand both are free.

So you have nothing to worry about, then. If you were running your own Matrix server, and wanted to use their push infrastructure.. Well you're still fine for now but who knows in the future.