Obvious to note that he had severe mental problems, so logic might not always have been his strong side... ironical for a fantastic mathematician.

And then the act of his wife cooking for him also entailed her pre-tasting it before serving it to him, so without the only person he trusted to do that he could hardly use ingredients to cook for himself, could he?

This is a pretty classic symptom of extreme anxiety disorders like obsessive compulsive disorder. The sufferer either only eats food that they have prepared themselves or only eats food prepared by a specific person (such as the wife in Goedel's case). It could also possibly be a manifestation of some kind of paranoia.

It's a bit dangerous to try to back-diagnose psychological disorders in people from the past, especially based on only one symptom, so please don't take the above as a specific diagnosis of OCD. If you would like to learn more about Goedel, I would suggest the mathematician John W Dawson Jr.'s book Logical Dilemmas: The Life and Work of Kurt Goedel. Dawson is the person who first organized Goedel's papers.