What about non-elite groups? (...) they are likely to be heavily influenced by the cognitive elite, especially in the longer run.

I think they are likely to be influenced by whom they consider high-status. If you succeed to make a conspiracy theory website seem like an authoritative source (it must seem as professional as the mainstream media, except that it brings "news you will not hear elsewhere"), that's all you need.

Why would anyone make a professionally looking conspiracy theory website? Aren't "professionality" and "crackpot thinking" kinda opposed in real life? Yeah, the genuine crackpots usually also have low regards for mainstream design or marketing. But a professionally looking conspiracy theory website is a great vehicle for political propaganda. So a foreign government may spend a lot of money and professional work to make a high-status conspiracy website, where the conspiracies are filtered, and only those that are neutral or convenient for the owner are published.

This is not a hypothetical scenario. Russians already have such website with online radio in my country. I see advertisements for it almost everywhere. The content is more or less: "Things your government wants to keep secret from you: Vaccination causes autism. West is the source of all evil. Russia is a paradise. Being a member of European Union is bad for you, because they will kidnap and abuse your children!" And of course, the website is "independent and alternative".