Some don't work at all, bc they must be GSF-only I guess (installed via Aurora store). Others like Whatsapp or Signal sort of work but sometimes the app seems to not be running (even though I set up the no battery optimization). And I get multiple unread messages at once when manually starting the app.

Same, they work fine and are immediate. The only exception has been Signal. Must be that I'm using sandboxed Google Play Services?

I wonder if they installed something like "adblockers" in their devices on networks that constantly kills keep-alive connection to Google servers which actually delivers the message.

Or somehow broke the device-side wakeup for that keepalive.