It assumes the LLM only runs once, i.e. it doesn't account for chain of thought, which makes the program not memoryless.

There is no such assumption. You can run/sample the LLM & the equivalent Markov chain as many times as you want & the logical analysis remains the same b/c the extensional equivalence between the LLM & Markov chain has nothing to do w/ how many times the trajectories are sampled from each one.