You're not wrong - "the staff ate it later" is a word-for-word translation, so it's kind of weird to leave out 美味しく. (among other things a meaningful translation would say "crew" instead of staff)

But the nuance of the JP here is that it's using a polite set phrase, not describing whether people enjoyed the food or not. A bit like how "a good time was had by all" is used to wrap up a story, not really to describe what kind of time people had.

tl;dr, 美味しく is there because the JP would sound weirdly flat without it, and you're right that "enjoyed" would probably be a better.