I think even "actual cognition" is highly unlikely to need anything much more than associative memories plus some state and a loop. The expense being having to "execute" a large number of steps rather than "just" effectively pulling learned results from "cache".

To be very reductive, an associative memory can hold a truth table. Put minimal state, IO and a loop around that and you have a universal Turing machine. Which is why the "it's just memory" or "it's just Markov chains" is so tedious - it says near nothing about the computational power of the system including the model.

There's plenty of reason, of course, to question to what extent we know the abilities of the models, but when people assume dismissing it as "memory" or "just statistics" or "just Markov chains" I usually take it as a signal they don't understand how few limitations that imposes.