טבח may also be translated as a chef.
While cook is the more accurate translation seeing as the word chef exists as שף, it is correct nontheless.
A cook and a chef are two different things. Anyone can be a cook.
A cooker and a cook are a bit different as well. A cooker has sometimes four legs and not everybody can be a cooker in a pub.
Ha'tabach lokeach achrayut al ha'tafrit.
Is it wrong: "responsibility ON the menu"?
"On the menu" would not be the way Americans would say it.
I would translate it as "over the menu." It's close to what you're saying and goes well with the other definitions for על, such as "upon" and "on."
Example: I lay a cloth over the table. (could use "upon" or "on" as well)