I don't often use IATA codes when talking about airports verbally, but when writing they are a lot simpler.
But yeah, the vernacular often wins out. Here in DC, you can tell a local by how they reference DCA. The official name is "Ronald Reagan Washington National Airport," but most locals call it "National" or "DCA".
Most folks call Baltimore/Washington International Thurgood Marshall Airport simply "BWI" because the official name is word salad.
Dulles International Airport is typically just called "Dulles," much like Londoners would use "Heathrow" or "Gatwick."
Los Angeles International Airport is almost always called "LAX" by basically everybody I encounter.
As mentioned above: in New York City, JFK is commonly used, while the other two big airports are "La Guardia" and "Newark" (though I know quite a few in the Tri State area who say "EwwwwwR" for the latter, given its not-so-stellar reputation).
I still know folks in and around Denver who call the "new" Denver International Airport (DEN) airport "DIA," which was its temporary IATA code before Stapleton International Airport (the original DEN) was shuttered in 1995.
Many Portlanders (Oregon ones) call their airport "PDX."