Double-spacing doesn't create trapped white space. Otherwise, any kind of leading would be considered trapped white space.
The reason why most instructors tell their students to double-space their essay (aside from being dictated by the MLA guide) is precisely because it increases readability. Lack of sufficient leading causes text to become cramped and taxing to read. And in fact, most layouts on the web don't provide sufficient leading.
A teacher may have to grade hundreds of essays a week. So it's important that the essays turned in are double-spaced to reduce the strain on the reader's eyes. You may find a single-spaced essay to be more visually pleasing, but you're not the one who has to read 30-40 essays a night.
This is a case where design aesthetics definitely should not trump usability/accessibility.
Note: I used to sometimes cheat and use 1.8, 1.75 or 1.5 spacing to fit a really long essay into the page count limit of an assignment (or just to avoid wasting another sheet of paper for a single widowed line), but I really wouldn't recommend it.
Personally, I think traditional typed reports, research papers, academic papers, etc. have their own pleasing aesthetic. It's not like a slick corporate brochure or an artsy web design, but they have their own romantic aura to them.
If you want to make your paper look nice and stand out in a good way, try purchasing some nice high quality stationery and type the paper using an old-fashioned typewriter, and hand it in in a nice file folder (get a string-tie one for added class).