> Before the days of doze mode & other battery optimizations, you could just listen & block on a socket, then let the phone go to sleep. Incoming 3G packets would wake up the phone, you grab a wakelock, then start doing things. From what I remember, at least a while ago Facebook Messenger did this using MQTT. But this is not possible any more.

It's not a coincidence that the Facebook app was known for being an absolute battery killer. Go to any android forum post about battery life from a year or more ago and you'll see that the first suggestion is always "have you tried removing Facebook?"