The overall point feels strained to me.
Yes, formal mathematics has such limits. We can't expect machines to be perfect and provably perfect. But the industry isn't assuming that. Why would it? Natural intelligence is not perfect or provably perfect, either.
Rather than certainty, measurement is often enough. We can't prove a program will always halt, but we can check it halts in a specific execution.
Approximation is also often all we need. Even if we can't prove that we can train a network with more than 50% success, if we can get multiple shots at that (using different data, or initial random weights, or training techniques, or something else), then we can reduce that danger exponentially. (I don't know that we have a guarantee of succeeding there, but this would be the hope, and I am not aware of anything showing it is impossible, unlike perfect provability.)
Finally, it is possible that perfect provability does work on the problems we care about. Godel and Turing etc.'s proofs rely on finding rare situations where we can't prove things - cleverly-constructed pathological cases - but perhaps human behavior does not fall into that set. Human behavior may not be a pathological case for proofs or learnability.