Interesting, this was the big 'a-ha' moment in the grad course I took on the subject. Surgery was clearly important too, but this seemed to be more apparent from Hamilton's work. The W functional was what provided the control needed. Also, calling it 'easy' feels like it dismisses the insights necessary to have developed the functional in the first place.
Happy to be corrected on this; I wasn't an active mathematician at the time, so everything I know comes from other accounts.
I called it the easiest part of his papers, not easy. Either way, it's actually not relevant to the proof. For example, I believe that Morgan and Tian's 500 page exposition of the proof doesn't mention it even once.
Happy to be corrected on this; I wasn't an active mathematician at the time, so everything I know comes from other accounts.