Jason,
The videos don’t really answer the question about the test I suggest doing.
What you need to look at the the LED behaviour when the stick is not moved anywhere, ie centered.
So, if you move it , once , to full X travel (left or right) to activate the LED, then leave it centered, the LED should flash with short flashes. The question is, if you then move the stick very sligtly off center, against spring pressure, can you get the LED to flash with even shorter flashes? The shortest flashes should be within the "play" zone in the center of the travel.
Then repeat the above for the Y axis.
Andy