I stand corrected, reading an interview with
Sullenberger confirms that being in normal law didn't help them, because they didn't reach the limits that would have triggered the envelope protections. But he felt that having them available was a good thing, he said about the protections "I had that as an ace in the hole".
https://www.historynet.com/sully-speaks-out.htm
So we stayed in normal law—all the flight envelope protections were intact. That helped provide essentially guardrails to prevent us from exceeding certain extremes in terms of speed or attitude, for example. So I had that as an ace in the hole. It turns out we didn’t need those protections; we never got to the limits at which they would have protected us from ourselves.
So, not sure why the NTSB felt that starting the APU was so critical for the outcome.