I don't understand why the Internet is not government controlled. Would you leave road maintenance to private companies? How about air traffic, or sewers or water? The Internet is as important, if not more important (in a Constitutional way) than these services, but we leave it in the hands of self-interested parties.
The difference between these and the internet is right of way and externalities. That is, the government is involved in roads and sewers because (1) you can't have a road network in most places without the ability to forcibly purchase small strips of land from lots of people and (2) until recently, there was no efficient way to charge usage fees in a reasonable way. The internet doesn't really have that problem: charging is quite simple and the flexibility of routing means we're not nearly so beholden to any one set of property owners.
Now, I think there might be a case made for making last mile internet either municipally run or a heavily regulated business in the same way that water/power/sewer service in cities is, but that's a very different argument.
Ideally, the last mile fiber would be divorced from the IP connectivity to the rest of the world, so that a residence would pay the same money to use that last mile fiber whether they're running 1 megabit or 1 gigabit. If you then had real competition for options from your central office to the rest of the world, you'd see performance for a given price increase dramatically.
This is basically how things are done in Sweden, at least outside the major cities. A municipality builds and runs fiber to every home, residents choose from a number of different ISPs and each ISP pays a flat rate to the municipality for each active subscriber they have. ISPs can then offer any type of service within reasonable limits. This means that I can get for example 100 Mbps service and VoIP for ~25 dollars a month, or 10 Mbps for 15.
I think that you're exactly wrong on this. It makes sense for government to build roads because, for one thing, it's physically infeasible to have multiple sets of road-builders competing with each other. There is finite land.
On the other hand, it's quite feasible to have many competing internet providers. I'd say we have too few, actually, and that government makes it worse. The airways should be more open and whatever infrastructure the public subsidizes (telephone wires, etc) should be equally available to all ISPs.
More competition is the key here, not a government monopoly. All we need the government for is ground rules to make the game fair.
It certainly is more so in some countries like China and North Korea and look how that's going. Government's aren't always benevolent, they have their own agendas too that may not match some people's expectations of online freedom for one thing, and it's not like government's don't also increasingly have business interests in part influencing them. There are a lot of lobbyists these days...