Put a loop around a Markov chain where you provide a 'tape interface' taking instructions from and feeding input back to, the state, and you have a Markov decision process with a hard-wired decision maker acting as the tape. Provide the right Markov chain, and you have a universal Turing machine. So the extension needed from a Markov chain to something that could be programmed to do what you describe - say by running an ML model - is only very slight. And we do provide loops, and state when we run inference, just not infinite.
I'm agreeing with your overall point, to be clear - my point is that calling something a Markov chain is effectively calling it trivially extendable to something that can in principle compute everything any physical entity confined to the known laws of physics can, and so what it boils down to is whether or not the model is trained in a way that gives it those abilities, and not the put-down of the potential ability of such a system that people usually intend the "just a Markov chain" as.
I'm agreeing with your overall point, to be clear - my point is that calling something a Markov chain is effectively calling it trivially extendable to something that can in principle compute everything any physical entity confined to the known laws of physics can, and so what it boils down to is whether or not the model is trained in a way that gives it those abilities, and not the put-down of the potential ability of such a system that people usually intend the "just a Markov chain" as.