It can't "lose" what it never had. :P A fictional character has a mind to the same extent that it has a gallbladder.

if the writings of humans who have lost their minds (and dialogue of characters who have lost their minds) were entirely missing from the LLM’s training set, would the LLM still output text like this?

I think should distinguish between concepts like "repetitive outputs" or "lots of low-confidence predictions the lead to more low-confidence predictions" versus "text similar to what humans have written that correlates to those situations."

To answer the question: No. If an LLM was trained on only weather-forecasts or stock-market numbers, it obviously wouldn't contain text of despair.

However, it might still generate "crazed" numeric outputs. Not because a hidden mind is suffering from Kierkegaardian existential anguish, but because the predictive model is cycling through some kind of strange attactor [0] which is neither the intended behavior nor totally random.

So the text we see probably represents the kind of things humans write which fall into a similar band, relative to other human writings.

[0] https://en.wikipedia.org/wiki/Attractor