1. Most teachers are willing to help out students. Why would a student pay for your service?
2. An obvious answer to the above is "we'll focus on the students that have teachers that are unwilling to help". From my experience, that's a small market.
3. You also have to consider that the ones who would be interested in this service are ones who really want to have a good essay (I'll come back to this later); the union of the two sets is even smaller. I'm not entirely sure you would be able to churn profits from just this subset.
4. Moving beyond teachers, how will you compete with EssayJudge? They offer an amazing service for nothing. The quality of their service is hard to beat already, and their price impossible to overcome.
5. How will you vet the quality of your "teachers"? If a student isn't happy with the response, will you refund the money? What if your customers are immoral? (I've found that the union of immoral people and students is quite large.) Looking on HN, I see somebody mentioning "chargebacks"; that would indeed be a big problem.
6. What is your target market? Are you aiming for people who want to buy ghostwriters, or are you aiming for people who simply want to get a good grade (from "Doing well on essays is part of doing well in school, and doing well in school leads to a better life!", that seems to be your target audience), or are you aiming for people who really want to improve their writing? It seems that you're focusing on the penultimate, but is that really a good decision? Why wouldn't they just hire a ghostwriter instead?
7. An easy answer is "they're pretty good writers already, they just need that little bit of editing". If so, it looks like you're focusing on the last group. That's where the money lies, but your advertising doesn't seem to be targeted towards them.
This is by no means a complete list of the problems you'll face. You should probably have answers to many of these before starting/going forwards though.