Yes, the Apple II also uses a resistor-capacitor timing circuit to determine the paddle position. The big difference, however, is that the Apple II uses the timer and a counter to produce a paddle position 0-255. Software can then use this value to e.g. draw a paddle at a position on the screen.

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.