As an American English speaker, I would never say "cooks brilliantly" - it sounds very British to me. I can't think of a good translation that both fits grammatically and honors the meaning of świetnie as I understand it. The closest thing I can think of is "Her husband is a great cook." I'm curious if anyone has a suggestion for a close translation that sounds natural in American English.
I am a British English speaker and I've never heard anyone hear say "cooks brilliantly". I agree with you that the most natural thing that people would say in everday life but which also honours the meaning is "Her husband is a great cook". Similarly, we don't say "his son cooks badly". Instead we would say "his son is a bad cook." Nevertheless in order to learn świetnie, żle etc, it may be necessary to compromise a bit. I hope my Polish will be as good as Jellei's English one day!!
"cooks greatly" is not accepted. "cooks great" is often considered ungrammatical, but actually, it technically is correct; "great" isn't just an adjective. Oxford (I didn't check any others yet) also defines it as an adverb, albeit with an "informal" tag. It still rather grates on my ear, but since it's one of the most common ways to say something like this and is technically correct, we have to accept it.