The pilots did not train for a specific root cause of a fault. They trained for a symptom (uncommanded nose down), and the procedure for that situation was unchanged.
> extra burden to quickly determine if something else is wrong
This isn't an extra burden. Pilots aren't doing root cause analysis for failures while they are responding to them. They are trained to try actions in a specific order until something works. It's not like the checklist used to have one action on it and now it has two - the solution in this case was a standard action on the standard checklist.
Pilots do not look around, say "ah, electrical short in elevator actuator" or "ah, bad angle of attack sensor" and then take a single action.
Unexpected situations are always extra cognitive burden.
From my perspective, it is a human factors issue that Boeing failed to consider.
Yes, the emergency procedure remained the same, but even well-trained pilots are still people. And changing the behavior of the system in such situations was not well-advised.
And you are writing this after many, many accidents that root cause was pilot not exactly knowing what or why something happens with a plane or plane autopilot (ie. AF 447)
Being apologetic of cost cutting on safety issues is dumb as it erodes culture of safety and encourages others to skimp on safety.