To be SEVIS approved the Flight School has to first be a FAA Part 141 School. That shows that they meet certain criteria laid down by the FAA including having fully audited training plan for every course that they wish to undertake. The FAA also make visits to ensure compliance etc. As a result, the Student is given certain privileges in terms of a reduced number of hours to attain a certificate etc. Also a student can immediately on completion of a FAA private cert- go immediately into a Part 141 Instrument rating without the need for the pre-req flight hours that would be required under the lesser Part 61 course etc.
Once a School has Part 141 approval - they can if they wish then go and apply to the USCIS for SEVIS approval which allows them to train overseas schools who need a visa for Flight Training typically a M1 visa. Although there are a small number of Flight "Academies" who area able to sponsor a F1 visa.
Getting Part 141 and then later getting SEVIS approval costs a Flight School many of thousands of $$ in time, effort, training of Instructors etc and FAA auditing. That's why some Schools don't go to the effort.
Long story short - if you are a foreigner and wishing to train for a FAA cert then you really need a M1 visa. Some Flight Schools may tell you otherwise but if you are 'caught' and found to be in breach of your immigration status you are responsible not the Flight School.
The TSA also require that you specify the visa type/number when applying for TSA clearance and they also provide the statement on their own website that if you are later found to be in breach, they will terminate the TSA approval etc.
To be honest - for the sake of $200-$300 for a visa - why take the risk and why go to a non SEVIS school who have not proven themselves able to meet the requirements?
How long were you planning to be in the US for this training and what training were you considering? Take a look at EASA, EFT etc - they all provide the ability to handle the visa paperwork. (This is not a recommendation for these Schools) - plus you can get a JAA license in parallel. Especially with the new Euro regs indicating a JAA license will be mandatory for Euro citizens.