The engine that failed didn't do so entirely, by the sound of it. It was still running, albeit not producing much thrust. When they mistakenly pulled back the power on #1, it would have been showing lower numbers than the dud #2, which reinforced their ideas as to which had issues. The engine identification should have happened with the thrust levers still matched. They never formally identified which engine. And when it became obvious that the aircraft wasn't going to fly, they should have pushed both levers up. Even an engine that is on fire, is producing thrust, and if you need it, then that overrides any other issues.
TransAsia Airways 235 comes to mind too.
Not all pilots are equal!
Does it need a quick diagnosis? Not really. Very few things need people to rush. The engine is unlikely to end up less dead because you hurried the shutdown.