The State owns USPS and it's not so bad. Before phones were introduced, it was the main channel of communication.
You can, of course, leave off the return address, and this works fine for letter mail.
Packages, however, require ID to be dropped off. You cannot use the USPS to send packages anonymously.
The analogous version for a state-run ISP would be a blanket filter/ban on Tor/VPN usage.
I don’t think the USPS is a good example of a state-run organization for that reason. The state has already demonstrated their desire to eradicate personal privacy and anonymity.
Others would take the USPS as proof that a strongman president can switch such services on and off at a whim, for political reasons.
When was it done?
I am supportive of municipal telecom infrastructure but this example always gives me pause.