I mean, why even bother to run flights at all in this scenario?

They could cancel 80% of flights and keep the rest to pretend they are still an airline.

Cancelations would be more profitable than the flights themselves.

80% would be way too much, the consumers would catch on and probably not buy tickets anymore. But don't worry, the airlines' best MBAs will be hard at work calculating the exact percentage of flights they have to fly before it starts hurting the bottom line. And once all airlines start doing it, they could bring that percentage down - what are the consumers gonna do if that's the only way to get to the destination?

They still have to offer replacement flights, and if the replacement flight isn't reasonable they have to refund. They can't just keep your money.

Don't forget that a lot of flights are business flights. Fortune 500 companies will negotiate deals with the airlines, and they will ensure that getting there matters. Sure the CEO flys the company jet, but the next level down rarely does, but they talk to the CEO and will ensure that the chosen airline will get their people there by contract (wherever there is), if the airlines start failing to get people there on time these contracts will change since the large companies have enough money to matter. Those who fly a lot (again likely for business, even small businesses sometimes have someone flying several times a week) again are people the airlines need to make happy as they will go to different airline if there are problems.

Which is to say they can screw the "common man" who rarely flies, but most of the business is people who have enough power to to to airlines that treat them well and at that point it normally isn't worth screwing anyone.

I imagine the class action lawsuits at that point would bankrupt them