If we're playing that game, there's a million things they could do as well, but it doesn't make any of them likely. Hell, they could do what you're talking about
right now, other countries do that even though their Internet access isn't controlled by the government. The porn ban in the UK is a great example of privately-controlled ISPs being forced to block certain sites.
I'll repeat, there is absolutely nothing about publicly funded Internet service that in any way even remotely connects "internet access as a public utility" to "government starts blocking certain sites". They could do it right now if they wanted to.