If it enhances one's writing skills then yes, I think it should be a required course. But there are other ways of developing writing skills without linking it to literature.
That being said, teaching writing in a literature course is still the best way to teach it. There is a close link between reading in writing. Seeing how others write improves one's own writing skills in terms of structure, diction, grammar, and so on.
Furthermore, it's worthwhile to improve one's ability to read and interpret texts. It enahnces one's thinking skills to be able to analyze text, be able to articulate its meaning, and come up with a thesis about what a text says.
Finally, just because one can read doesn't mean one can read everything equally well. Some writing is better than others, and some texts are more challenging than others. I hated "The Canterbury Tales", for example, but agonizing my way through it was worthwhile in the long run.
Hmm - I hate using the "you'll be better off for it" argument. Too nebulous. So let's try it this way: I really enjoy reading and literature. And I get more out of reading, and out of a variety of kinds of reading, because I have at times tackled different texts of varying difficulty. It also helps to read statements in the news, and see what peoples' assumptions are and where a person (such as a politician) is fudging.