As someone with very limited single engine experience long long ago I'd love to get some thoughts from those in the pointy end. Just looking at the track between 23:25 and 23:29 I'm slightly confused. My understanding is that if the correct memory items where followed you would have established pitch and power as constants...normally in a gentle climb while troubleshooting. During those 4 minutes it appears that both speed and altitude remain "relatively" stable but with clear oscillations that appear to indicate an inverse speed to altitude shift so there is clearly an issue maintaining level flight. I've always thought that at higher altitudes hand flying a swept wing plane was significantly harder than at lower levels. Under an emergency would the aircraft commander normally take the controls and rely on the FO to trouble shoot or would he be more likely to allow the FO to maintain level flight while he looked at the issues?
It would appear that there was an upset and recovery at roughly 23:29:30 and then again roughly 2 minutes later that proved unrecoverable. In both instances it appears to be a roughly 800 ft drop over roughly 20 seconds...that seems awful "ham handed" for purely pitch induced control input to me without some sort of rolling oscillation involved??? What type of flight path variation would be "normal" for an experienced airline pilot with minimal hand flying currency? Hard to imagine losing the plane at 5000 ft. in benign VFR conditions without some other issue in play???