EGOL,
Firstly, I want to state what a truly great answer this is. Since most people are not playing in this niche, they may not appreciate the insights you've pointed out in such little time.
Content is going to be a large part of our strategy. While others have done this to an extent, you'll notice the same bland, boring stuff recycled over and over. There's a huge opportunity in this space to take content to a new level and create truly great stuff. That's what we're planning on doing.
Regarding creating multiple service pages for these specific keywords... I think you're absolutely correct. When I can't figure things out, I tend to think obsessively about them until I believe I have an answer. Last night, before heading to a friend's birthday party (and before you answered this question), I reconsidered my standpoint (also after I poured through some Adwords search and conversion data for our site.
I decided that the best approach would be to create a general "essay editing" page with footer links to pages like "fix my essay", "revise my essay", etc. Like you, I realized that there actually is different intent for these search terms than there is for "essay editing service" and I greatly appreciate your answer in helping confirm my hunch.
The one place I do believe you're not quite spot on is where you say, _"This is not an easy business niche. Your competitors have been around long enough that they will not be defeated quickly or with little effort." _ This niche has not been disrupted like many niches have in the recent years. The players, in my opinion, have become complacent and nobody has been attempting to become an industry leader here (I believe there's definitely an opportunity to do so). The guys ranking toward the top of the SERPS do not have very impressive link profiles, they don't create good content (if any), their user experiences and design of their websites are awful.
I'm not saying it won't take effort and time to surpass them, but I do believe it's one of the more "doable" niches out there.
Again, thanks for the great answer. If you've got any more insights, I'd love to hear your thoughts.
Cheers!