It's not the technology design that's that important. It's the network effect, and peoples' default trust in megacorps over volunteer projects. Both of which cannot be solved with just a protocol.
I think the natural and equivalent role of the USPS would be an ISP, rather than a "messaging platform" itself.
When the US Constitution was drafted in 1787, authorizing the new Federal government to run a postal service, carrying letters and packages via horse rider/wagons was the state-of-the-art.
It was always relatively trivial to roll your own messaging service. And open protocols exist (and predate messenger)! The thing you can't (easily) replicate are the network effects.