The next step beyond the mail was the telegraph and then the telephone. Those were built out by private companies, but operated under a regulatory regime called "common carrier," which protected the network owner from liability so long as it charged and carried traffic from all parties equally.
These days, that concept is called "net neutrality" in the context of the Internet, and I suppose you may have heard it mentioned in the news recently once or twice. (I think it was a mistake to rename common carrier, but too late I guess.)
Personally, I think that private buildouts are a better way to go than a government-funded network infrastructure. Private industry is better at investment and product development than the government, and any enterprise that generates a profit contributes to the long-term economic sustainability of innovation. A government-run economy becomes starved for capital and innovation slows way down; that is why the Soviet Union lagged U.S. technology and why Cubans still drive 50-year-old cars.
But privately run networks need regulation to deliver the results that citizens want. Railroads needed it, telephone companies needed it, airlines needed it, and yes, ISPs need it. And if that is thwarted, then sure, let municipalities take a swing at it. My town is looking into muni fiber now, and I'm in favor of it because the local service from Comcast is so bad and so expensive and there seems to be no other way to do something about it.