Even while Trump's idiot was in charge, tampering with the mail was a federal offense.
And I don't imagine it being totally public. USPS would own the network, and companies would pay them to sell access. This would allow dozens of ISPs, all offering different levels of service, without needing local monopolies. It would look a lot like dial up ISPs: You could use AOL, EarthLink, NetZero, or any of a handful of small, local ISPs.
The USPS would then contract with companies to maintain the networks, but since they own it they can regulate the ISPs a lot more closely on a host of issues. Plus it would give free access to all online government services.