Your regular mom and pop flight school operating under Part 61 cannot themselves be SEVIS approved.
However, the TSA website will happily take your money for training in a Part 61 school - as I discovered
Despite the fact that you will never get a Visa to go there.
There is nothing to stop the SEVIS approved Part 141 school from conducting Part 61 Flight Training for the foreign student.
Which is how practically all U.S. based "PPL/IR" stuff is done - you don't do a Part 141 course. You go to a Part 141 school only to get the I-20.
It is a pity that Part 61 schools cannot do foreign students.
Well, they can, because you don't need a Visa for certain classes of training.
The 141 school in AZ I went to was definitely a "mom and pop" outfit
Worked just fine.