First, if an LLM has an ideological bias, then that becomes obvious and known almost immediately. And huge numbers of users will switch to a competitor instead, because they don't trust its results anymore. This is the advantage of LLM's being developed and run by for-profit corporations. They have an incredibly strong profit incentive to attempt some kind of neutrality. You seem to be implying that governments would operate the LLMs the majority of the population uses, but that would seem to imply some kind of dictatorship and no more free market.
Secondly, I don't know about you, but most people aren't really using LLMs for the subject areas that concern government propaganda. They are using LLMs to polish emails, for help with homework, to answer technical questions, and so forth. Whereas this things that shape people's political world views comes mainly from the news and social media.
You seem to be envisioning some kind of a world where people don't access the news or social media directly, but it is somehow passed through some kind of LLM transformation filter. I'm not sure why people would sign up for anything like that. If I see a link to a New York Times story, I want to read the story directly. I don't want an LLM to rewrite it for me. And I don't know anybody else who wants that either. Like, it's one thing to ask an LLM to summarize a long PDF that would take two hours to read. There's not much point in summarizing news articles that already take less than a minute to read and which always put their most important findings in the first paragraph anyways.