I think you make a good point but lack the linguistic facility to differentiate the salient point. Which is not your fault btw but that of the language you are using
German makes a distinction between art (Kunst) and “artistic crafts” (Kunsthandwerk). The line between the two is somewhat fluid but still pretty clear. Kunsthandwerk is an artistic creation that has a purpose other than a decorative one. It’s made by people who consider themselves craftspeople first (and not artists) and it is made to be sold (which art is sometimes as well, but not always).
There is still a distinction between normal crafts and Kunsthandwerk, though. You have to imbue your work with a certain aesthetic sensibility and a desire to do decorative work above and beyond the purely functional.
I would argue that cooking can be craft and it can be Kunsthandwerk but not Kunst.
So it depends on how you want to translate this statement into English whether you consider cooking an art or not.
Sidenote for the pedants: yes, it could be argued that the Arts and Crafts movement did make a similar distinction. Their understanding of craft certainly included a decorative element. But as far as I know they never tried to make a distinction between artistic crafts and normal crafts, probably because their main point was that all crafts should have an artistic element. I would go so far as to argue that the clumsy name of their movement shows that they were linguistically constrained in developing their ideas
