My fps is pretty bad with the add-on airports. When I did the demo flights, I made the big mistake of using FSX aircraft and everything was fine with fps. After reading this post, I did your test. With my add-on aircraft, the fps is very jerky, but when I tried an FSX default aircraft, it was fine.
You haven't done any mistake. In fact, if with "demo flights", you mean you were testing JFK in Trial mode, that was the ONLY correct way of testing the SCENERY, which is with a default airplane. If you test a SCENERY with lots of AI and an fps-heave airplane, you lose the ability to judge the scenery itself, because if the fps is slow, you don't have any means to understand what is causing it.
Now that you made that test, and got proof your low fps wasn't caused by JFK, you might try to better understand how to improve your situation.
My last message said that you surely must upgrade the video card, if you see an immediate fps improvement, if you go down from 4096 to 1024 textures, without changing anything else.
But if your fps stays low (with your add-on airplane and everything else) even with the 1024 textures size, chances are that changing the video card (unless yours is really outdated or, even worse, it's integrated in the mainboard, like the Intel HD graphics), won't change much, and you might have to rethink your entire system.
But the cheapest upgrade of all, it's just not using products not well optimized. The worse offenders are those still using FS8/FS9 code which is slow in FSX, such as World of AI traffic, for example.