They had to react quickly, that's why experience counts, they had not much time to guess the proper sequence of actions. Here in a forum we have all the time in the world...
And that's exactly why I said guess
Ok, going back to ET302:
The only way a pilot would have significant experience with solving this problem is if they had been so incompetent to begin with that they had screwed up the trim way out of whack.* They would have gotten this experience by allowing a plane to become dangerously close to losing control... probably with a load of passengers on board. This grossly out of trim problem is so rare, they moved the diagnosis to a supplemental reference and deleted the solution, entirely, some decades ago. I suppose they figured anyone who would need the deleted info should really just not fly planes. Since the pilot is now experiencing it for the first time, with an intermittent runaway trim on top, and they don't have "all the time in the world" the way you do, do you think it's at least possible that Capt Sully and his experienced copilot could also have botched it?
*It's not like the pilots have many/any opportunities to intentionally pull stunts with their employers' 737's, even on the rare occasion they are flying a near empty plane with just a few other crew.
Also, the reason they undercorrected is perhaps not so strange. Every pilot is fully aware of the dangers of overcorrecting in this scenario. Some pilots are apparently unaware, at this point in time, of any significant danger of undercorrecting, thinking they can just eventually get the trim into adjustment in stages, without being aware of the other parameters which must be considered. Hopefully this is no longer the case.
I don't see any obvious reason Sully would have known any better. He was an experienced commercial passenger pilot, not a stunt pilot. This is like thinking an experienced bus driver will know what to do when the bus goes into a slide. We pick bus drivers that don't slide the bus in the first place. This trim problem is so far out of the regular "lines" it doesn't normally exist. And if we only let experienced Indy car racers drive buses, I don't think it would make travelling by bus any safer. Personality type and temperament might increase the accident rate.
GoJ: we've been around and around, so I will also say this:
The memory item says nothing about looking at the trim indicator and putting it back to green band by using the trim button before powering it off. It says the button CAN be used to correct trim before stab cutout. That makes it sound like it's optional. And that the "correct" amount of trim is just a judgement call which is not necessarily of vital importance but more for convenience of the pilot. Maybe the pilot has some other additional training/education that covers this stuff in more detail, and perhaps that additional education and training should have been adequate. But the emergency procedure should perhaps better cover the actual emergency and not just the malfunction that can lead to this emergency in the first place. Going back to Jakarta, the captain did not appear to realize how close he was to losing control of the plane. Like you, he thought, well, just press the up button. Give control to the copilot and start reading a manual.