Are you saying that we should essentially cut out everything such that the entire track ball unit fits flush against the mounting plate on the underside?
Yes.
This would mean we would ... cut out the space necessary for the entire track ball unit so that there is nothing between the unit and the mounting plate, and the whole assembly is attached to the control panel via the four screws in the mounting plate?
Basically yes. The reason it was working without cutting all the way through with your mockout was the cardboard is ~1/8" thick. However, wood (+plexi) CPs are between 5/8" - 1" thick; usually 3/4" wood plus (if any) plexi.
the dumb question ... why not just cut that 3.25" square area around the ball?
To address this directly, the ball is in the case, and the case must be very close to the top of the CP for the ball to stick out above the CP. Here's side views of the tops of two different happs models:

The left is the "high lip" model, the right is the old school aka "normal" aka "original".
As you can see, with the normal & high lip models, it is possible to cut the hole smaller than the football to the size & shape of the case just below the lip, but you'll have to route the football shape so that arm doesn't get stopped by the bottom of the CP (assuming your CP is thicker that the ~0.5 inch difference). Note that there is a third model, the "high ball" model, that you cannot cut+route the underside like this, because the body of the case is the thickness of the arm.