That's assuming the consequent: you are assuming that such an enumeration strategy is requisite for strong AI to occur. [EDIT: I misunderstood what you meant by enumeration, my statement of your assumption is wrong. What you do assume is that in order to be able to outperform humans at any task a Strong AI must not be tripped by diagonalization. Or that humans would not be any such output, that is that the human can somehow step outside your conundrum.]
Yet it works just as well to consider humans to be char programs which map strings to strings. But then, they somehow manage exhibit Strong AI. Actually I believe Scott Aaronson has used this table look up type argument to show how you could theoretically pass the Turing Test.
Now
imagine a lookup table that stores every possible history H of A and B’s conversation, and next to
H, the action fB (H) that B would take next given that history. Of course, like Borges’ Library
of Babel, the lookup table would consist almost entirely of meaningless nonsense, and it would also
be much too large to fit inside the observed universe. But all that matters for us is that the lookup
table would be finite, by the assumption that there is a finite upper bound on the conversation
length. This implies that the function fB is computable (indeed, it can be recognized by a finite
automaton!). From these simple considerations, we conclude that if there is a fundamental obstacle
to computers passing the Turing Test, then it is not to be found in computability theory.
But you are going in the right direction. More developed versions of your argument lead to why a Bayes Optimal AI is not possible.
Note also that as Strong AI is a vague term I have defaulted to the nearest solid metric: prove it is conscious.