Let this be his issue. If he is awesome, he will do well no matter where he goes.
Have him apply to the State U. Tell him you just want him to have one application in so he has a place to go. They admit mostly on SAT/ACT and GPA. If he doesn't want to apply to other private schools that is fine.
But! Tell him to imagine it is April 1, and other students are hearing if they got into various schools. Ask him if he would wish he got into a certain school? or at least had the chance?
Talk to the GC and see if the English teachers have an assignment to work on essays.
Don't put pressure on him to go to a Top 20s. Tell him you want him to have the options he wants..whatever it is. "If you don't want to apply to Dartmouth, then that is fine. If that is why you don't want to do the essay..no issue."
Ask him to schedule a time to work on essays..." I know you have homecoming and Joey is having a party soon...when do you think you could finish these essays?
Realize your son is doing what every other kid is doing.