Some people with an English literature or English writing degree are effective teachers of English as a second language, but many are not. If I were hiring second-language teachers of English in an overseas country (as I have had some occasion to do in previous foreign residence), I would look for someone who was a native speaker of English, yes, but preferably someone who majored IN SOME OTHER LANGUAGE so that I know the would-be teacher understands the task of a language learner.

There are also masters-level professional programs in teaching English as a second language, which are usually quite well designed.