I dug up a couple photos in my email of the ball drive Eric was talking about, we were both on the team in 2005 when it was developed in the offseason. As Eric mentioned, it shared a lot of conceptual similarities with yours - 4 corner balls in a square frame, driven by 4 motors between each ball. It was also inspired by the 2-ball omni drive that Technocats built in 2003.
These were taken when we first assembled everything, as a first pass fit/function check of the captive ball. The lego tires shown on the rollers are just placeholders, and we later had to switch to a stronger hinge at the pivot point above the springs.
We tried several different balls to get something to work - we wanted something in the 4-5" diameter range. Many balls we tried were not quite round enough or too rough (dimpled pitching machine softballs). What is pictured are actually garden gazing balls. They were cheap and light, but welded together from two halves, so they had a bit of an oblong shape. The springs helped accommodate this. Also to Eric’s point about preventing driving the balls out from under the chassis, our outer corner ball transfer unit was placed below centerline of the ball (much like your latest update) to mitigate this.
Some of the biggest functional issues stem from the contact point between the ball and the driving rollers/wheels. To drive, you need friction here to transmit torque to the ball, necessitating a fairly significant normal force on the roller, and a grippy material. When the ball is being driven by the opposite roller, you want the ball to spin freely with low friction about the roller contact point. Finding the right balance here can be tricky.
I do like the use of larger wheels as a single roller contacting two balls. It is simpler (less moving parts) than the belt/roller system we had, and there should be minimal load on the wheel shaft, as the normal forces from the balls on either side are balanced. The layout works pretty well because of your choice of larger, basketball-sized balls.
I’ll try and do some more digging. Somewhere I have a low quality video of ours driving down a hallway.