I suppose it's been a question I've asked myself a lot over the past three years: does giving extended due dates and more time actually lead to better work? In short, I don't really know. I can say that it doesn't lead to worse work. Averages grade-wise over the past three years have been marginally better, but it's difficult to tell if students do better than they would have done otherwise. To measure how a student stacks up against their own hypothetical self poses problems in a possible research study. As a result, I'm forced to rely on anecdotal evidence. I can't compare the early-takers with the late-takers because it's likely the early takers are already good at English prior to even taking the course, so naturally their numbers may be higher.
In the end, perhaps it doesn't matter. All I can do as a teacher is give opportunities and feedback. Everything else has to come from them. At the end of the day, they'll have to decide how much effort they want to put into the essay, and how much "waiting" is actually going to help them.