It seems to me that LLMs do not keep on thinking, "I'd better get this right, or I'm going to lose credibility." They have a theory of mind, but it stops there at simply having one. It's not like they're thinking about the 2nd and 3rd order implications.

To put this into perspective: Imagine interacting with another person, who doesn't value Truth at all. Or perhaps remember an occasion when such an interaction happened. In general people don't like these interactions, and they react with distrust and even hostility towards such people.