I don't even know where to begin to address your confusion. Without computability theory there are no computers, no operating systems, no networks, no compilers, and no high level frameworks for "AI".
That is patently false - most of those things are firmly in the realm of engineering, especially these days. Mathematics is good for grounding intuition though. But why is this relevant to the OP?
I'm not equating anything here, just pointing out that the fact that AI runs in software isn't a knockdown argument against anything. And computability theory certainly has nothing useful to say in that regard.