Is कि used to mark a quoted clause (in this case, "the food is bad")?
Yes, it's equivalent to 'that' in English:
These people say THAT (कि) the food is bad.
It can also introduce non quoted clauses:
मुझे लगता है कि खाना बुरा होगा।
I think that the food will be bad.
Another word borrowed from Persian. In Persian it is که (ke).
Can the complementizer, कि, equivalent to 'that', be omitted in Hindi like it can be in English or Swedish? (In Romance languages like Spanish or French, it is typically obligatory; in Chinese there generally is no complementizer.)
No,it cant be omitted. It is used almost all the time while speaking.
Im not sure in the technical space but colloquial norms dont require it.
This sentence is confusing because it can also translate to These people say that eating is bad.
The correct sentence should be "...यह खाना बुरा है।" The यह means this to denote this food rather than just food or eating.
What is these people's mode of nutrition?