Duh! That makes perfect sense.
Make sure its the same 5V supply line for both pots. If its separate lines - might not work - although you said it worked with an external power supply, so I don't see why it wouldn't.
Power and ground are just jumpered between pots, so no worry there.
I'm sure this concept will work. It shouldn't require anything but a few bucks worth of resistors and diodes.
The diodes will be needed to prevent voltage from going backward through the other gear circuits since all of them will be wired to the same two Logitech wires.
Now I'm trying to decide whether it makes more sense to:
have a set of resistors for all six gears + neutral followed by diodes
or
only have sets of resistors for the three values for left/right and the two values for up/down, but that would also require diodes before the resistors because each switch is going to be connected to two sets. This might end up using just as many parts, just not duplicated parts.
The first one is probably the way to go, but need to diagram it to see if there are ways to simplify.
Not sure when I'll get to it. In the mood to work on more mechanical things today.
If someone else wants to draw up a diagram, have at it!
EDIT: Here's the concept with just two gears. The final design would have six, but another circuit for neutral. The power for neutral would be daisychained through all the NC terminals of the switches, so that if no switches were pressed, the correct voltages for neutral would be sent. A couple gears will require three resistors to get the correct voltages.
(I'm a noob at schematics if it doesn't show)