In my experience the grounding wires don't usually have much effect.
This is usually true for contact and hall effect switches, but capacitive switches are completely different. For example, xwhatsit's controller will not even work unless the ground clip is properly attached.
Note that his is working better without it attached.
I've run XTs and ATs fine without the ground wire attached, without the case. I suspect it might be more vulnerable to having a problem without it in the rare case, but that has not been my experience.
Note that it's because the shield return is not attached and the grounding wire is.
Both of those keyboards will not operate at all without the ground wire. Also, in both cases, the plastic case / metal bottom pan have no electrical function of the keyboard.
I just got out my AT model F and opened the box and the case and measured it and the backplate is attached to the shield clip via a standoff (as it should be). The backplate is also attached to the electrical ground of the controller, so it can properly shield the capacitance from stray Electro Magnetic Interference.
If this were not the case (that is: if the backplate of the KB were not electrically tied to the electrical ground of the controller) there would be issues.
I bet if you disconnect the controller from the plate and disconnect the clip it won't work as well.