This is a good example of when the simple present tense (I cook) would be acceptable in English. From what I understand, both the simple present tense and the present continuous tense (I am cooking) translate to the same verb form in Russian. In English, we use the present continuous tense (I am cooking) to talk about things we are doing right now. The example, "I used to cook pork. Now I cook chicken," uses now for contrast, rather than to indicate we are currently engaged in cooking.
"I cook chicken" is rather awkward in English. Unless you're telling someone about a specific way that you prepare it" For example "I cook chicken with salt and pepper" or "This is how I cook chicken" "Right now I'm cooking chicken" would be the most natural way to translate it.
I agree, it does sound a bit awkward, but on other Duolingo languages they allow a bit of flexibility between I cook/I am cooking (for example).
Unless there's an actual grammatical reason why this would be false in Russian, I wonder whether both shouldn't be accepted.