I'm still confused that you have a plane in the meat of it's flight envelope with known attitude and power settings in what should be a stable climb and it all goes to hell within seconds?
The plane is not in a complex environment in a highly automated state where the pilots focus is elsewhere and they are relying on the automatics and have degraded scan and over reliance. It's not AF where the senior pilot was off the flight deck, the least experienced had the controls and poor CRM prevented the more experienced pilot from intervening while he might have been able to recover...
A qualified pilot flying a functioning aircraft manually during a "routine" (in the sense that it was briefed and trained and not an emergency) lost control during what should have been a simple pitch and power maneuver that could be flown my almost any general aviation pilot with suitable training.
By that I mean that if you took a typical 500 hour GA pilot and gave him 40 hours of 737 sim training and 20 hours of manual circuit flying (take off, landings, approach and departure patterns, touch and go circuits etc.)
He would be able to pass an aborted ILS approach at any point it was called by a check airman. Or is this a wrong assumption?