This would be a natural role for the Post Office to take on, to provide a neutral ad-free, privacy-respecting messaging platform accessible to all.

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.