Explore, if you intend to do a full-time course (which is 18 hours or more of practical work per week) you are not allowed into the US under the visa waiver program. So you need an M-1 visa for which you need the I-20 form plus some other paperwork
The visa needs to be "sponsored" by a flight school and for this to work the flight school needs to be SEVIS approved. There are two types of schools: part 61 and part 141 (or something like that) and only one of those types is SEVIS approved by default.
Theoretically you can wiggle your way out of this if the course is part-time and not the main objective of your stay in the US. But that's a rare situation and most officials (including the friendly INS official that decides whether you're admitted into the US or not) will expect you to have an M-1. So it's probably you who will have to explain/prove that you don't need a visa for what you're attempting.
In addition to this, I understand that the TSA (which needs to approve things in your case anyway) checks for the M-1 visa and will ask questions if you don't have one.
Sooo.... Good luck! Let me know if things worked out as planned.