"If it were to periodically "check in" with an external server to see if it needs to do any kind of spying -- admins would notice the network traffic."

Admins would use network dedicated hardware with network chipsets driven by closed source firmware. In a crazy but technically doable scenario, that closed source firmware could contain instructions to "send packets containing magic word X to address a.b.c.d" and not reporting or counting that traffic, even to applications opening the device in promiscuous mode, with all routers in between to obey the same instructions. Not a single byte would be reported, counted or sniffed unless someone sticks a digital analyzer on the network cable.