> Matrix doesn't even support server-to-client message pushing, only polling. Which means, if you want your phone to receive a message while your client is in the background, you have to use a non-Matrix protocol to receive those.

(aside long polling which I cannot comment) why is this a even an issue?

Centralized push services (may it be FCM or unified push) are at this point well established.

As a user using a specific messenger app as a push service doesn't gain me anything, it only makes it more difficult to switch messenger app if i choose to use it and mostly use an additional push service (hurting battery life), because most receive push notifications via Google anyway.