I don’t necessarily disagree with your interpretation, but there’s a revealed preference thing going on.
The number of non-tech ppl I’ve heard directly reference ChatGPT now is absolutely shocking.
The problem is that a lot of those people will take ChatGPT output at face value. They are wholly unaware that of its inaccuracies or that it hallucinates. I've seen it too many times in the relatively short amount of time that ChatGPT has been around.
To avoid the darker topics to keep the conversation on the rails, if there were a misinformation campaign that was trying to state that the Earth's sky is red, then the fine tuning should be able to inform that this is clearly fake so when quoting this it should be stated as incorrect information that is out there. This kind of development should be how we can clean up the fake, but nope, we're seemingly quite happy at accepting it. At least that's how your question comes off to me.
So the labels thing is something that obviously will never work. But the system has all of the information it needs to know if the question is definitively answerable. If it is not, do not phrase the response definitively. At this point, I'd be happy if it responded to "Is 1+1 = 2?" with a wish washy answer like, "Most people would agree that 1+1 = 2", and if it wanted to say "in base 10, that is the correct answer. however, in base 2, the 1+1 = 10" would also be acceptable. Fake it till you make it is not the solution here.
You can kill people with a fork, it doesn't mean you should legally be allowed to own a nuclear bomb "because it's just the same". The problem always come from scale and accessibility
How to enforce it is the real sticky wicket though, so it's only something best discussed at places like this or while sitting around chatting while consuming