Ok, well this is the circuit sketched-out in a way that might look a bit more textbook. It's a non-inverting amplifier circuit with an offset. The gain is around 5x.

The general gist is it amplifies the hell out of the signal going into it, so the pot can't be moving much to go full scale in the game.
If you simulate a sweep of all possible voltages from a 5V pot (resistance doesn't matter), only about the central 20% does anything (between the yellow markers).

I can't see what electrical difference having a 5k pot makes to a 1k pot.
Going off that thread on A.P., it just looks more like the two types of sticks have different ranges of movement, so if the Gundam stick doesn't have the right movement around that sweet spot in the middle (it sounds like it has way more movement), folk are frigging things with resistors (I must say, hacking up the amp board seems a bit extreme if you can put resistors in-line with the stick, or just do away with the amp board).
I suspect if you have the monkeyball stick but run it without an amp, it's working because of the adjustable calibration in the game, but it's going to be getting a much smaller voltage range into the game (20%), so it might be more prone to noise/jittering.
That's what I think anyway, happy to learn more if I've messed up.
LTspice file attached for anyone who wants to muck around.