You are right that often degrees are required for jobs that don't seem to need that kind of qualification but I'm a little confused by this quote. In my experience unless you are an English major university students rarely take required English courses, maybe one in first year. When they say you don't have to take another English class to earn a degree I think they mean they are satisfied you are not a low level ESL student or somehow challenged that you can't function at a basic level in a regular class. I don't think it means you have some sort of high English level that his higher than the average university graduate.