Demand and supply is a basic concept of economics. High demand and low supply equals higher prices. For a manufacturer there can be nothing better than a perception of scarcity about a product they are making; a limited edition, short time only, one off deal. Better get it while you can.
Elite private schools use this perception of scarcity to their advantage too. Hell, their entire business model is built around their clients desire be part of an exclusive club. They limit the available places, even though they have many more people willing to pay for their services. They tell parents that there is a position for their child - but only right now, and it might not be there next year! Better take this opportunity while you can.
Why? Because the scarce resource is desirable. The scarce resource is prestigious. Being part of the exclusive club gives you status.
Tell me why then, in this day and age, of modern communication technology - why can't anyone go to Yale? Or Harvard? Can't I watch the lectures online, submit the assignments digitally, take part in tutorial groups and discussion via Discord or some other platform? Why then can't I take the course? I'm willing to pay.
Because it doesn’t benefit an elite private school to make their service available to everyone. They'd lose their status, they'd lose the appeal of being exclusive. The demand might increase but so would supply, and therefore, the price (financial and metaphorical) goes down. The value of a 'Harvard' education decreases, their accountability increases, as now they must justify their education practice to the masses instead of the self-selected, elite who would have paid double for it anyway, just for the prestige.
A school that spreads the myth of scarcity isn't a school, it's a business.