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.
Moreover I'd strongly dispute that there was any particular point where it was clear that the problem was "near a solution." The W functional is an example where experts could very quickly verify that Perelman had made at least one major new discovery about the Ricci flow. But the proof of the Poincaré conjecture was on another order of complexity and required different (more complicated) new results about Ricci flow.
Moreover I'd strongly dispute that there was any particular point where it was clear that the problem was "near a solution." The W functional is an example where experts could very quickly verify that Perelman had made at least one major new discovery about the Ricci flow. But the proof of the Poincaré conjecture was on another order of complexity and required different (more complicated) new results about Ricci flow.