Or, in a deeper aspect: they're pretty good at regurgitating what we interpret as bullshit. They simply don't care about truth value of the statements at all.

That's part of the problem - you can't prevent them from doing it, it's like trying to drain the ocean with a small bucket. They shouldn't be used as a direct source of info for anything that you won't check afterwards; at least in kitnaht's use case if the LLM is bullshitting it should be obvious, but go past that and you'll have a hard time.