But only if necessary. In most cases the "meat" either implied ("What kind of meat is that?" "Pig") or replaced by the specific name of the cut (Schweineschnitzel, Putenbrust, Lammkotlett).

The "extra" english words succinctly add context then.

They only exist because nobles wanted to pretend that the food they eat is so much fancier than that of the peasants, so they insisted on using French to describe it.

Assuming that this is true, that also adds a distinct meaning to the word so that saying "pork" as opposed to "pig meat" would have served as an indication of class status in addition to e.g. what you'll be eating for dinner.

I think a better proof would be “chicken” in English. If we get by without the distinction for one animal, it doesn’t make sense that we’d need it for another.