I'm thinking more about instructors owning their own planes, based at a big GA airfield and offering their own freelance instruction, unaffiliated with any flying school.
OK, let's take a hypothetical situation. Fred the Flying Instructor decides to do this. He turns up at a big GA airfield...but they reckon they have enough schools anyway, and he doesn't want that much competition. Ditto for airfields 2 and 3. He eventually finds somewhere. He rents a portacabin and buys - or leases - an aircraft. He jumps through the required CAA hoops. He gets a couple of students and all is well for a little while.
Oh dear, G-XXXX needs to go off for an Annual. Fred tells his students it won't be avaialble for a bit, but they're not happy. Fearing losing them to the flying school at the other end of the airfield, he leases another aircraft - it's a bit tatty, but what can he do. Maybe he can get rid of it later on.
But no, he can't. Because either it's raining and all aircraft are grounded, or all his students want to fly at the same time! Anyway, a couple are near getting their PPL now, and want to hire from where they learned to fly, and if they can't...there's the other flying school at the end of the airfield, so maybe they'll finish the course there too. And actually, Fred can't safely teach and supervise solo students at the same time on the few sunny days he gets, so he really needs to employ another FI. And on a sunny day when they're flying their little socks off, that's the time when everyone turns up and wants to know about flying, so he really needs someone in the office. And if he doesn't sort this all out in a professional manner, and have instructors and aircraft available when people want them...they go to...you've guessed it...the school at the other end.
And suddenly..................Fred is running a Flying School.