I agree. I usually start with the first paragraph, but if I am unable to complete the first paragraph (where I state my stand), I just go into the body. I usually have the outline on a piece of paper, and probably erase the whole word doc a few times. On this topic: I don't like setting an upper bound for word limit. The longest essay I have ever written was 15 pages because I had so much to write...
The key is getting a draft out, literally a raw simple draft of what you're gonna say, a week ahead of time. It's kind've like pseudo code. Then fill that in.
Then I'd hammer on my papers over the week. They'd start out raw and get better and better. I'd tweak. I'd find a better word. (Is this sounding familiar?) I'd work on it.
When I turned it in, I knew I was getting an A. I had two papers quoted in lecture. Hey wait, I wrote that! I don't think I ever got less than a B+, generally A's.
Writing isn't magic. It's work. In fact, what 'they' do isn't really that much different from what 'we' do.