There does seem to be an increasing correlation with adherence to right wing politics and cog-loving. Another reason is likely the tendency of right wingers to autopmatically simp for oligarchs and everything they do. So in their minds using AI is "owning the libs".

The biggest problem is that when something is absolutely factually known to be true- like "Vaccines save lives"- It doesn't matter. If one were to push it hard enough, it will eventually cave and agree with them and call whatever cockamamie theory got them to 'vaccines actually cause autism and are used to covertly implant mind-control chips in people's arms and will render the entire population of white people sterile...*'

they use LLMs because they sound submissive and subservient. But they quote it because it agrees with them.

(*this is an actual set of beliefs two of my uncles held. One of them died from COVID, though he expressly refused an autopsy in his will, so he will appear in one of the 'excess deaths' statistics. The other lost a grandson to COVID.)