"The teacher does not speak English."
The Chinese sentence, 老师不说英语·, could indeed mean, "The teacher does not speak English." The exercise provides no context; so, there is no way to tell from the sentence as presented. Maybe the teacher is an American spy, and she does not speak English because she knows that doing so might blow her cover and reveal her true identity.
Yes, 老师不会说英语 would indeed clearly mean that the "teacher can not speak English" in the sense that the teacher does not know how to speak English, but this exercise sentence is simply 老师不说英语; "the teacher 'can not' speak English" is one possible interpretation, and may well be the reason why the teacher "does not," but we do not know that for certain.
Remember that in Chinese, the distinction between habitual present and present progressive is not as strict as it is in English. Accordingly, 老师不说英语 could also simply mean "the teacher is not speaking English" (right now).
A better way of writing that in Chinese would be "这位老师不会说英语." While Chinese is all about reducing complex concepts into the shortest phrases possible, as the sentence currently stands, it's awkward because we don't know which or how many teacher(s) (since both singular and plural form of teacher is 老师) and whether or not s/he is choosing not to speak English or does not know how to speak it.