Saying something should be run by the government because it's important seems like backwards logic to me. Should the government manage the production and distribution of food as well? Should all housing be government housing projects? Basic food and shelter are far more important than internet access.
The US government pours tens of billions into food production.[0] Direct subsidies account for 15% of agribusiness ("farm") revenues and the US government both underwrites and pays 63% of the costs of crop price insurance.
Every developed country subsidizes their food industry to some extent or other. In the EU, the Common Agricultural Policy is the largest item in the EU budget.
Food subsidies seem to be economically essential for a stable food supply, and politically essential to ensure stability in rural areas.
Shelter, for its part, is heavily subsidized through the GSE[1] system. The GSEs underwrite mortgages, lowering interest rates for homebuyers, and stabilize the mortgage system by allowing mortgage issuing banks to offload the costs of internal failures onto the public instead of locking up the mortgage lending system.
Tighter government regulation of internet availability, including public or quasi-public ownership, would be far from exceptional given the degree of government involvement in other essential economic areas including food and housing.
I don't see a meaningful distinction between "control" and the existence of selective subsidies. The US government directs agribusiness to produce specific crops by giving the industry more money to produce those crops congress wants. This is "control." Food safety rules are also "control" of the market.
As far as housing is concerned, GSE preferences (combined with additional support like the mortgage interest tax credit) distort the housing market by supporting owner-occupied housing, which is abnormal by global standards. This is deliberate shaping of the market, or "control" by most definitions.
Also with regard to housing, the market is directly controlled through zoning laws and building codes. Zoning controls housing availability to artificially inflate prices, and building codes theoretically control construction quality.
The government does not have a monopoly on the production and distribution of food or housing. They may use taxpayer dollars to induce supply in this respect or that, but they are not the sole provider.
