While I'm sure most profits/margin come from first/business class, I think that's only viable because the large number of economic class passengers help marginalize the cost per passenger. To give a concrete example, suppose the airline is flying 20 business seats per flight. The cost of those seats to the airline if it had no economy class, and it were forced to allocate an entire plane for them would be immense -- marginal the cost per seat would be (illustration) ~Airplane_Cost/20+Accomodation_Cost. Now suppose a current airline will carry perhaps 200 economy class passengers where it makes 0 profit. The marginal cost of the business class seats to the airline now will be much lower, something like ~Airplane Cost/(200+20)+Accomodation_Cost.
So the airline don't need economy class to make profit, they need them to create economies of scale that bring down their cost for the high margin customers.
I have been on a 747 that was business class only (Singapore Airlines). But the economy class passengers are definitely required for scale on most flights.
So the airline don't need economy class to make profit, they need them to create economies of scale that bring down their cost for the high margin customers.