Of course, what's missing from hypothetical artificial arithmetic is simple symbolic manipulation. And I am pretty sure that "More gofi", a more explicit treatment of AI, is something that Noam Chomsky and other have as their favorite bullet point on how to go further.

However, that too has been tried to whatever degree. If you had an overt, tractable logical theory that explained all intelligent behavior, then yes you'd have the missing ingredient of human intelligence. But unlike arithmetic calculation, it seems unlikely that human intelligence has this quality.

Indeed, all human "dealing with the world" behavior together involves a black box whose entirety is not subject to rational or logical reflection or description but whose broad outline. A person can't give a complete-enough-to-write-an-algorthim account of walking down the street recognizing things but the person can likely give a good why the street light they see is a street light - ie, heuristic black box behavior and logical/deductive behavior is intimately tied within human behavior.

And this might give some clue what's missing modern AI.