What are you doing about the missing GCM/FCM? Especially with messaging apps it's super annoying when they only receive updates when you actually open them.
Sorry for the late response: email (K-9 Mail) uses IMAP idle, Signal uses a persistent websocket, and XMPP (Conversations) also uses long-lived TCP connections. I don't use any proprietary messaging apps that rely on Google's push notifications.
As someone who has been doing this for a while, I find sometimes I don't get a notification until I open an app. You get used to it. Anyway, when someone calls me on Whatsapp it still works fine for instance.