This is now about Erdogan, Does Erdogan need NATO more than NATO needs Turkey? That's the question. Dictators tend not to care or stay rational in the "normal" sense of things.
If this coup was planned by Erdogan, then he really has good game and will play a long con on being a member of NATO while doing what he wants.
Well, except Iraq.
That is, there is considerable tension between Iraqi Kurdistan and the central government (the fact that Kurdistan has its own army and the Iraqi army is banned from Kurdistan is one of the irritants on the Iraqi side -- the US arming the Kurds in a way that would make them able to effectively resist Turkey would not be a good thing for the internal stability of Iraq. Or for Iraqi-Kurdistan relations if it happened after Kurdistan broke away completely, like it has shown some interest in doing recently.)
(Actually, reinforcing Iraqi Kurdistan -- and even moreso an ex-Iraqi Kurdistan -- this way would also escalate tensions with Iran, Syria -- and because of the last -- probably also Russia.)
Refugee handling is an EU problem, not really NATO. While there's substantial overlap, the US isn't going to negotiate with that in mind. And there's nothing stopping the EU from continuing to cut a refugee-handling-for-money deal if Turkey exits NATO.
Airbases can still be leased, although it would depend on how badly relations deteriorated.
I think Turkeys biggest fear would be that NATO members stop quietly ignoring Turkey's bombing of PKK bases inside Iraqi Kurdistan and Syria. If Turkey wasn't in NATO, there's nothing stopping the US or others from giving various Kurdish groups weaponry which could threaten Turkey's air superiority (AA missiles, not jets). This would make Turkey's war against the PKK internally a lot more difficult.
So, I do think Turkey will recognize that it will suffer greatly outside NATO, and will take at least basic steps to preserve membership. But we'll see.