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.