How idiomatic is "He cooks badly" vs. "He doesn't cook well" & "He cooks poorly"?
All three forms appear to be equally idiomatic but the last is not used often.
It must taken into account that there are quite a few false positives in the result for "cooks poorly".
(ref. 1, 1992) Whoever cooks according to rules other than the proper ones for cooking, cooks poorly. But whoever follows rules other than those for chess plays another game;
(ref. 2, 2002) Gradually he adjusts to Aunt Joan who cleans obsessively and cooks poorly. He even realizes that his wimpy cousin Wesley isn't all that bad,
(ref. 3, 1989) Damaged cassava cooks poorly and often tastes unpleasant; tannins may also bind to proteins in the gut, preventing their absorption. (Notice that here, "to cook" means "undergo the process of being cooked")
(ref. 4, 1979) A woman who cooks poorly is “ a woman of heavy bread ” . A person ' s character is also described in terms of bread
It is possible that when "to cook" means "undergo the process of being cooked" the adverb "poorly" should be preferred, but that is just a flitting impression of mine, I can't verify that as a fact.