@Anonymouse - If you use a bit the size of the smallest diameter of the hole, you could do one side, then the other (think venn-diagram in how the drilling would overlap).
I tried this mod and here are my observations:
1) Very nice.
2) There is enough material to deepen the hole a bit on the replacement, OR
3) When putting the controller back together, tighten the three screws on the side of the analog stick to finger tight, then back them off a quarter turn. That will make the joystick move freely so no modification to either the replacement stick or the original controller is needed and the controller will still hold together enough that you shouldn't notice the screws aren't torqued all the way.
Note: This replacement (or, at least the one I received) is not a perfect match. The dome portion extends down further on the original than the replacement, so when you move the stick to its fullest extent in any direction, there will be a gap that allows you to see inside of the controller.