User experience

Not sure, isn’t that to be expected? Doesn’t Matrix use those corporate push service providers to notify the mobile devices, and that would require you to register your server with them?

Using an XMPP protocol server (e.g. freedombox’s ejabberd server) should support to work with direct connections to the clients (can be tested with https://compliance.conversations.im).

More matrix protocol drawbacks:
FBx day 4: Matrix Synapse's howto
It was too easy, so now what? - #2 by NickA