OK. I certainly don't want to contribute to mis-information or mislead anyone to the extent that they get in to trouble with the US authorities, heaven forbid, but...
A good friend, James Dunn, CFI at long established Cessna accelerated flight school
Northaire in Prescott, Arizona and an FAA examiner himself told me the following in person just last Saturday:
"The M1 Visa for flight training is history and applying for it just complicates matters. The new TSA rule supercedes the visa process. The TSA screening requires the applicant to register by internet and does not involve any delay when arriving for flight training which can proceed while the application is being checked by TSA. In Northaire's case the actual fingerprinting is carried out by the local Sherriff. It would only be an issue if the person was rejected by TSA, at which point flight training would have to stop. There's no question of a person pitching up for flight training subsequently being reported for a visa violation"
James is not a charlatan and the school I understand is approved to issue the I-20 form. I suggest that anyone considering this route check with their selected flight school and get confirmation similar to what I have quoted above from the examiner who will carry out their flight test.