Was the miracle on the Hudson saved exclusively by the APU?
Yes, the solution to start APU was important.

The ditching procedure directs the use of maximum available slats and flaps for the final approach and touchdown (source, chapter 10.3). This is not possible without APU, as Airbus A320-214 cannot move flaps if only powered by RAT (only blue hydraulic line, same source, chapter 9.3). The running APU adds the green hydraulic line that can also move flaps.

Hence, without running APU, the optimal ditching procedure is not possible; that gives less chances for success.