An associative memory can model truth tables of logic circuits, and so the steps to go from associative memory to arbitrary computation pretty much requires adding state and a loop. And we do provide state and a loop. As such, while that does not prove that any given model is capable of reasoning, describing what goes on at any step as associative memory says nothing meaningful about the computational power of the wider system.

It's very possible the way we train current models will turn out to place fundamental limits on the abilities the models will have, but that will not be because the models act as associative memories.