What I understand is that the ESTA program is essentially aimed at speeding up the VWP processing, but is not a replacement for the VWP.
I might be wrong, but
AFAIK here's the procedure.
1. At home, you need to fill in your details in the ESTA system. After a few automated checks, you are granted authorization to *travel to the US*. But this does NOT in any way grant you permission to *enter the US*. You still have to apply for the VWP at the border. (Without authorization to travel to the US, I think there's even a system in place that ensures that you are refused boarding by the airline. Not sure how this works though.)
2. At the border your VWP application is checked as usual, but the immigration officer now has the benefit of having your details electronically on file.
I honestly don't know what sort of checks ESTA does before giving you approval to travel. This might just be a very cursory "were all fields filled in properly" field to a detailed check to see whether your name appears on terrorist watchlist etc.
But in any case,
AFAIK, ESTA is NOT independent of the VWP. You have to be admitted under the VWP properly first, before you can fly into the US in a private aircraft.