You need to read the TSA ruling to see where the visa comes into play. It is repeated in a number of places but taking just one bit:
"The flight training provider must keep a record of the following for 5 years for each foreign applicant:
The photograph of the candidate;
A copy of the approval sent by the TSA confirming the candidate's eligibility for flight training ( this is the final determination sent to the candidate and flight training provider via e-mail after the candidate has applied for flight training with the TSA, paid the $130 fee, and submitted fingerprints);
Candidate's full name, gender, and date of birth;
Candidate's ID number created by the TSA;
Copy of candidate's valid, unexpired passport and visa ;
Copy of all previous passports and visas held by the candidate and all the information necessary to obtain a passport and visa ;
Candidate's country of birth, current country or countries of citizenship and each previous country of citizenship, if any;
Candidate's requested dates, type, and location of training;
Candidate's current US pilot certificate and certificate number, if any;
Candidate's current address and phone number and each address for the previous 5 years;
Copy of receipt confirming that the $130 was paid;
Flight training is defined in the rule as "flight training for the issuance of a new certificate or rating or the addition of a certificate or rating in an aircraft weighing 12,500 pounds or less". It does not apply to flight reviews, proficiency checks, or other flight training not for the purpose of a new certificate or rating. Nobody yet knows if it applies to training endorsements such as complex, high performance, tailwheel etc because the TSA have not yet decided.
It seems to me that unless you have a visa you cannot commence flight training for a new certificate or rating because you could not complete the TSA requirements at the very least. I also think that if you arrived at a US airport and told them you were in the USA for flight training and did not have a visa you would be refused entry. I suppose the only way to find out for sure is to turn up and try it but my sincere advice is not to be the first one to try!
The requirements to demonstrate you have a visa are repeated a number of times in the TSA rule.
I don't know if this will satisfy everyone but as I said before you can't come to my school without obtaining a visa. All of my students with a visa had no trouble entering the US which is really what this is all about isn't it?
Regards,
Chopperpilot 47