No, that's not correct for any reasonable definition of "impossible." Look up pangram's accuracy ratings. It's not perfect, but it's pretty good. LLMs in fact leave very distinguishing traces of their logit distributions in the text they write. It's one of the reasons why it's so easy for humans to also smell them.

It is possible to trick pangram - they bias toward a low false positive and a higher false negative - but it is not true that it is essentially random.