It's a nice hack. I think this is how Woz did the Apple II paddles, too (albeit with discrete components and a timer IC). It sounds familiar.
The game chip, on the other hand, doesn't produce a value. Instead, the timer is synchronized to the screen drawing, so when the timer fires, it indicates to draw the paddle now. So the system never has the paddle position as a digital value and there's no register that holds the paddle position.
I haven't figured out how the ball position is stored. It seems like they'd need a register for the X and Y position, along with an incrementer/decrementer, but I couldn't find that. Maybe there's something tricky going on. Or maybe I haven't looked in the right place.