The alps joystick came in today. The hack works as expected, but there are a couple issues that might prevent this particular joystick from being a good choice.
For Crossbred900 in particular, the small size might be an issue.
The joystick handle is about 14mm in diameter and 40mm from the top of the handle to the joystick base.
The handle does unscrew from the joystick shaft, but the threads that it goes onto are pretty small.
I guess the sanwa tops could be used if an adapter was machined for them.

The other issue is that there is a fair amount of play with the joystick centered, to the point that the springs do not push it all the way back to center. This is less of an issue on a PC with emulators because you can set a deadzone. I don't know how much of a deadzone an Xbox360 allows for. I will test this out on an Xbox 360, but I only know a couple people who own one and it will probably be after Thanksgiving before I can hit them up for assistance. I tested the little original pots out of the controller to make sure that there was no built in dead zone, but there was not. They are just plain pots.
The inside of the Alps joystick:

For anyone else doing this hack, the joystick and button that is depressed when the joystick is pressed down are one unit and both need to be desoldered before trying to remove it.

Here is some video of it working. You can see where the X axis has issues returning all the way to center.
Toward the end, I show how much play there is in the stick when it's centered.
So where we stand is that an analog joystick using 10k pots does work, but strong centering is probably going to be important.
I'm going to go ahead and hack the rest of the controller to barrier strips and hopefully not ruin it.

When I get access to an xbox 360, I'll turn the pots by hand and see how much of a deadzone the xbox is expecting.
Keep an eye out for other possible joysticks. They don't have to be 10k. We can swap 10k pots in.
It would be nice to find a used set of Happ analog sticks. I don't know if I've ever seen a used one.
Does anyone know if the bat-top joystick shaft from the analog Happ will drop in a Happ flight stick?