It would be expensive to build a new train to JFK. The unions and regulations in NYC make those projects very long and very expensive (look at the 2nd Ave subway line). There is an "AirTrain" to JFK but you have to take other trains to get to it first. There was supposed to be one to LGA but it got cancelled. We used to have a really nice water shuttle to LGA but that also stopped many years ago. People didn't want to travel to the water shuttle and pay $20 to get to the airport in 15 minutes. I'm hard pressed to see how a cheap quadcopter ride is going to be anything other than a novelty unless the FAA allows the heliports to be built inland -- we've had a bad history with blades flying through the streets.
Funny how every other developed country manages to build more infrastructure cheaper despite having stronger unions and stricter regulations.
I guarantee France have stronger unions and regulations, and still managed the GPE. 3 years late and with 20% cost overrun, sure, but to be fair, they had to deal with floods twice, which wasn't planned and broke equipment and reseted some tunnels.
I don't see how unions cause any of those problems. Corruption and incompetence comes through administration and management not the average worker wanting a decent pay and 2 weeks of vacation.
There's a free bus to LaGuardia from the subway.
One thing that some cities have done where awkward infrastructure is required to get a train to the airport is to, essentially, borrow money to do it, and make the fares to the airport very high to compensate.
Notably, getting to Brussels airport, which takes about 15 minutes from Brussels Nord, costs about 15 euro. For a 15 minute train journey. Hands-down the most expensive train per minute (or per km) I've ever been on. But, at least in theory, it's paying for this thing: https://en.wikipedia.org/wiki/Diabolo_project
(That's by no means the only one; lots of airports are in awkward places so running rail to them is expensive, and it's common for it to be paid for by special, more expensive services. And people use them.)