I don't think that intuition is entirely trustworthy here. The entire space is high-dimensional, true, but the structure of the subspace encompassing linguistically sensible sequences of tokens will necessarily be restricted and have some sort of structure. And within such subspaces there may occur some sort of sink or attractor. Proving that those don't exist in general seems highly nontrivial to me.

An intuitive argument against the claim could be made from the observation that people "jinx" eachother IRL every day, despite reality being vast, if you get what I mean.