> Why do you need to “check them” in the first place though?

I have this problem; most Android apps send background notifications via Google's Firebase Cloud Messaging instead of rolling their own services. (Signal is an exception to this.)

So, if you're unwilling to use Google Play services (or us an OS which doesn't support it), you're fucked and will not receive background notifications. The primary reason I use bridges with Matrix is not to unify my messaging experience into one app, it's to get notifications on my phone.