By the time air travel became economically accessible to the masses, such that high-classed public transportation to airports was on the political radar at all, New York City was already one of the most heavily built-up areas on the planet.
Fitting an entirely new transportation system into Lower Manhattan at that time would have been a non-starter, so the only way to provide a one-seat ride would be to integrate it with one of the transit systems already in place. The options are then LIRR and the subway, both of which were already working under congestion. This would severely limit the service frequency a new JFK line could get without reducing service for existing users of the system.
Since JFK is right next to the built-up area, getting a heavy-rail connection (with the attendant limits on gradients and curve radiuses) into the terminal area would also be difficult, possibly requiring politically troublesome demolitions. A peoplemover allows more flexible routing (and we'll come back to that). Digging tunnels under the airport might not have been feasible, given the low elevation (and proximity to vulnerable wetlands).
Finally, and possibly the kicker: With a large multi-terminal airport such as JFK, "single-seat train ride to the city center" is kind of an iffy proposition in the first place. You can't make the trains stop somewhere that is convenient for all the terminals, so many passengers would have to transfer between the train station and their terminals using some other mode. Perhaps a peoplemover, which can more easily snake around between the terminals?
(This is what they have at San Fransisco and Chicago O'Hare, for example: The city transit system does connect to the airport, but then most passengers have to change to a airport-internal peoplemover before they reach their check-in desk. Or, in Europe, consider CDG or London Gatwick).
And what JFK has is exactly that: A peoplemover that connects the individual terminals to the subway and LIRR. Giving one or perhaps two neighboring terminals a subway station would not make it appreciably easier to get to the others than it is today.