Reduce the number of flights to what can be comfortably served by existing infrastructure. Reduce maximum planes size. Make it a domestic terminal for flights to several major airports where large, intercontinental flights depart.
As I understand all the problems originate from too high traffic at various points and all of them would need to be expanded. So what if you instead reduced the traffic to the narrowest choke point size?
Do build a large airport away from the city. If someone goes on a 14 hour flight, an extra hour of commute won't change much, especially if it means comfortable and organized boarding experience. OTOH it makes a difference if you want to visit your aunt seven hours of driving away, if you can get there through one hour of commute, half-hour of boarding and an hour of flight.
Closing down all that existing infrastructure would be a huge waste of money. Smaller planes mean safer landing. Less traffic means safety in air. Fewer passengers mean the terminals, parking lots and public communication will be able to handle the numbers comfortably, without need for upgrades.
Sure the solution shares some disadvantages of leaving the airport running at current capacity, and some disadvantages of shutting it down and building a new one - but it seems to me it resolves the worst of disadvantages of both.