This is likely to be in the cards in the future, definitely. I don't think they would ever get rid of the internet outright, since that would mean depriving people of all their circuses, and they would ACTUALLY rebel against that rather than just sink into lower states of depression and isolation, like they are with covid.
I think what would likely happen is they would maybe shut it down for something like a week or so, just to give people a taste of the power the government has over their connections, entertainment, social media, etc, and how much people need the internet. Then turn it back on, but restricting it to maybe top major corporate websites, using some kind of justification that they have to prove their intrastructure is immune to cyberattack or whatever. Whatever sites do not make their whitelist cannot be accessed.
My question is how they would be able to enforce that? Would the point of attack be the Domain Name Servers? If someone has an ip address they can just plug into their browser, what could the PTB use to stop that?