I concur! If a bus driver goes on holiday, he'd have to take a bus to get there. So he doesn't really get away from his work. In your example sentence, the suggestion is that the husband-chef will be cooking for his family. So much for a break from work!

A busman's holiday in many respects is no holiday at all.

If you are a bus driver, then obviously, you drive buses. What if you're on a holiday from work and you end up taking the bus? That's a busman's holiday. The same for if this chef is on holiday - he will at some point probably end up cooking for his family.