The videos of JLC explanations have been edited, I guess for length. The audio clip on the audio page however is not edited and contains additional details I haven't seen anywhere else (8minutes and 29s long).
First of all when JLC saw KE in his life raft he was with two reefs, no head sails and not slow enough. He was close enough to tell KE he was coming back. He lost sight not because he started the engine, but because he had to take another reef. Then he came back 5 times to the site but failed to locate the raft in the decreasing light and huge waves.
He then continued to search, following the instructions from the race war room. At some point during the night CROSS Gris-Nez got a position from a beacon, transmitted to JLC via the race HQ. JLC turned around and went there, but didn't see anything. But a few minutes later he spotted a flash for a few seconds between two waves a mile and 1/2 away. He did follow the light, came very close with the intention of bumping the raft on the windward side, but despite going full reverse on the engine he missed it. With the raft slipping two meters behind the stern he threw the "banana" and its yellow polypro line (that he had prepared when he took the reef) and KE was able to catch it. It then took a lot of effort to bring the raft close enough for KE to climb aboard as the combination of the sea anchors on the life raft and Hubert drifting at 3 knots was not ideal.