I dont think you need an expensive pot, for a home use controller. Ive replaced arcade pots with Radio Shack pots. They probably wont fail for 5yrs of home use play. If it was on location.. thats a whole different story.
FYI - Star Wars actually auto-calibrates in-game. When you play your first game.. roll the controller around to its fullest travel points, at least two times. After that, the game should be calibrated, and will center the pointer in accordance with your yoke centering.
Can you snap some other pictures of the assembly at different angles? Im curious to see the x-axis, behind the plate.
As for a "Delay", there might be a controller response issue, if that sliding lever arm has gotten worn down.
As the driving-pin pushes the level arm up... its contacting the top surface of the sliding arm... When you push in the other direction, the pin has to then hit the bottom surface of the sliding arm. The problem here.. is if there is too much gap space between the sliding arm and the driving pin. If the arms surface, and or the driving pin, has worn down... the gap becomes larger.. and do when you change directions... you have a moment of control when nothing seems to happen. You have to wait for the pin to travel through that empty space, before it finally hits the lever arm.
To fix this, you may be able to have a machinist make you a new pin and or lever arm... or both.
As a temporary fix, you might be able tighten up the gap by making a straight edge that can attach to the side of the lever arm.. by using some epoxy paste, and a piece of wood (popsicle stick), strip of metal, such as an aluminum ruler, or a strip of plastic.
(Use a soft metal, because you dont want to wear down the driving pin any more)
If the driving pin is shot (no longer round).. you might be able to affix it to a vise.. then sand it with a fixed based dremel or drill press. Once sanded a bit, ...possibly putting a sleeve-ring around the thing. (Or just adjusting the arm to fit the new, now 'smoothly round', diameter)
This is the problem with lever arms, and why gears were the majority choice in arcade machines.
Once fixed.. it should last a very long time for home use. Just clean and lubricate it every so often, to reduce premature wear.