Like the river finds the sea, people will find a way around it. Satellite connections, just as an idea.
Anything a chip does can be backwards engineered to fool it. People will break your proposed surveillance chip eventually.
Most of these companies are maybe US-owned to varying degrees but they don't produce everything in the US. Also, they would put a very high price on these government mandated chips for two reasons: 1) government has deep pockets and 2) it would keep them away from very profitable so-called AI biz opportunities.
The pandy has shown us that with a few disruptions in the supply chain, any system that requires a cryptographic chip check to function can be sent to hell in a handbasket. I forgot if it was HP or Canon or some printer company had to teach its customers to bypass, i.e. hack their own cryptogtaphic chip checks because they couldn't get more chips and otherwise the printers wouldn't print. A few disruptions could also affect the censorship chip supply chain.
The great firewall of China has also shown how creative people get to get their message across. If it's not just human censors but also so-called AI censors it will just take creativity to a new level. Necessity is the mother of invention.
So there are some reasons why you might be worrying too much. I think another one is much broader. The majority of Americans did not vote for the current president. If he started censoring the internet now there would be Civil War II - Now It's Digital. The reason why Russia or North Korea can censor their people much easier is because they have never had or only on paper a brief period of liberty and rule of law. It will be much harder to control the US population. There isn't just the one media outlet, the one ISP, the one judiciary to dominate. It's splintered. And populated by feisty people, some of them armed. You couldn't pull off what you suggested without much more support for 47. And he seems to be losing it more than gaining these days.
Sci-fi is delightfully circumspect on how an intergalactic empire would work. Maybe Herbert's Dune universe is clearest and he just took us back to the middle ages with sandworms and drugs, fiefdoms and nobility.
I think whatever area shares the same government is a country. It doesn't have to be contiguous or on the same body floating through space. It could be the size of the Vatican or half the universe.
I suspect the definition of the word will change once (if) we make it to the stars. We have gone from nomadic life to loosely defined borders to kingdoms to empires to multinational and intranational federations of sort. These terms may no longer be fit for purpose when we colonize Mars etc. And maybe that's why you struggle to comprehend how it would all work behind the scenes. We don't know for sure, sci-fi authors don't know (or don't want to be too specific and limit themselves in what stories they could tell in the future).