As @const_void@lemmy.ml points out, there’s a bunch of players that can scrobble directly to listenbrainz.
But even if you use some player that does not have support for it, you can make any player that can scrobble to last.fm work with listenbrainz instead since they provide a compatible API. This includes even software which officially only supports last.fm by simply changing the scrobble destination it wants to scrobble to in your hosts file.
It really is pretty nice software.
Not all notifications go through FCM but all push notifications do as far as I’m aware - which is what the previous comment and the post title are talking about.
It is, in fact, worrying for privacy implications on the one hand and a real monopolizing factor on the other since if you wish to deliver an app which needs to implement such notifications you’re using Google’s service or constantly drain the user’s battery.
There’s UnifiedPush which tries to provide an open alternative but so far unfortunately still sees very little adoption.