The government are not going to let any old backyard Flight School go issuing visas to any Tom, Dick or Harry.
Again, why not?
The government does issue temporary work permits for any old backyard commercial corporation that wants to get a foreign Tom, Dick or Harry work at their place for a few years. Those companies do not have to go through a year-long, $2K+ review.
So what's so fundamentally different that the government simply does not allow I-20 visas (or an equivalent) for students that want to get trained at a Mom&Pop place?
And in any case it's not the flight school that issues the visa. It's the government that issues the visa once a flight school has acknowledged that the student is going to be training with them.