No because them pointing at a door is answering a different question than the one that was posited in the question.

no because the question was which door would the other person say is safe, they both point to the not safe door

The answer is stable because the liar will always say the bad door is safe and the truth teller will always say the safe door is safe, therefore the liar will always say that the truth teller will direct you to the danger door and the truth teller will tell you the same.

I tried to add some self-reference to the question to make a paradoxical answer but can't see a wording that even causes something like "this statement is false", at least not one about which door to pick.

Only ways I can think of start with the paradox right in the question. Like "If the other guard said, 'this statement is false', would you believe him?"

Sucks someone downvoted just for asking questions to better understand this less than straightforward thing. I've always believed that if you think something is wrong, you should challenge it, because even if you are wrong, the resulting discussion can help you understand why your previous perspective was flawed, which might then cascade to other things you didn't realize you were also mistaken about.

The question we ask if "What would the other guard say if I asked him which door is the good one?"

Liar says Bad Door

Truther says Bad Door

Now, for their answers to update, they would have to ne answering the question, "Which door would the other guard say if I asked him 'Which guard would the other guard say is the good door?'"

We want a guard to answer "What would the other guard say is the good door?" Regardless of how they answer our "outer" question, the answer to the "inner" question ("which is the good door?") doesn't change.

Liar doesn't care that Truther would say that "Liar would say the right door is the good one," Liar is being asked how Truther would answer "Which door is the good one".

I know I basically just said the same thing three times. My brain isn't working to break this out the elegant way I can't quite assemble. But hopefully some part of all this helps. The crux is that the question that they are imagining the other guard's answer to is not the same question they themselves are being asked.