> you don't want to prevent power-saving by keeping a TCP socket active

Slight aside: keeping a TCP connection consomes almost nothing. In fact, that's how push notifications work today on Android: a TCP connection is kept open, and apps' server send a ping to Google to tell the mobile to wake up.

What prevents power-saving is if every app keeps a different socket open to a different server and pings the servers on different schedules. That forces the mobile to wake up constantly.