Can you think of any immediate reasons this build would not work with an Arduino Uno? And if so, do you have any tips or recommendations for make the swap easier?
The Arduino Uno is not USB compatible. It has a serial to USB converter on the board, but it can not be programmed as a generic USB HID. So it can't be used in a joystick and be plug and play.
The hackery required to make it work isn't worth the effort in my opinion, as the pro micro can still be programmed in the Arduino IDE, and is just as cheap if not cheaper