It could be the driver but it is unlikely based upon this statement:
When assembling and disassembling the unit, the gated picture gets selected sporadically, but no matter what it always defaults to sequential.
It sounds like it is the switch itself or a loose wire. What happens if you press the switch manually while watching the properties page? The switch is right there on top where the plate attaches. (sounds like you know this already)
If it is a loose wire it should be simple enough to fix if you can solder. Take the thing apart far enough that you can see the underside of where that switch is. If the wire is loose it should be obvious.
As a side note... I know about, let's say a few hundred, people using these and I have heard about very few issues with them. The most notable was a customer that had issues with the USB connection on the bottom of the shifter.
If it's not a loose wire that you can easily just solder back on, or you don't have the means to do so, just have Thrustmaster replace it.
Best of luck.