What everybody seems to be forgetting is the simple logistics of this. I am not even talking about the politics.
Suppose that by some sort of miracle, or lack thereof, depending on your perspective, the necessary political and physical issues were overcome (cab signals, power compatibility, Harold layout, etc). Now you can run trains direct between Stamford and Jamaica. But, which trains do you run like that? You must realize that the ONLY people who would take them would be people traveling between Stamford/Lower Westchester and LI or the airport. NO ONE ELSE. No one would use it to get between Stamford and Manhattan (and don't say change at Jamaica!). This MIGHT work if all flights left at one time, and everyone who would want such a service would be traveling at once. But that is not the case. The airport is in use 24/7, and people would want this service 24/7. Unfortunately, not enough people would ever want to use it at any given time to make it feasible. And don't say to reroute existing Stamford-GCT trains there, as that would simply take trains away from where they are needed.
Also, it is kind of assinine to go to this much trouble when only ONE of MNR's THREE lines would benefit. You ask why there is no Stamford-Jamaica service. I could just as easily ask why there isn't any White Plains-Jamaice service!
Just wait a few years, then walk downstairs at GCT to transfer to a train to Jamaica!
A lot of people seem to think that they are entitled to have a one seat ride from home to wherever they want to go. Well, I am sorry, this is not the purpose of public transportation. The purpose is to move people along the routes that the masses demand (i.e. into and out of the city). And, as things exist today, you CAN get from anywhere on MNR to JFK, you just have to change trains a couple times. You say that you want to do it cheap, well, you currently can, but you have to pay for that in time and the use of your legs. If you don't like that, drive. If you don't have a car, well, I am sorry, but it is not the responsibility of the government to bend over backwards to get people specific places fast and cheaply.