And without going into detail, the correction from AoA for the static pressure is minimal (other corrections like ground effect, reverse thrust etc are more pronounced and still small).
It is not so much the amount of error, it is the computers detecting that there is a failure and therefore switching off certain functions, leaving the pilots with an unclear situation which they have to understand and to handle manually. Or even worse, it may trigger the computers to start certain functions, leaving the pilots even more surprised.
The stabilizer is an immense “secondary” flight control surface.
In normal transport aircraft operation I tend to understand it even as the primary pitch control. Looking at a typical flight, the ammount of elevator position changes is probably less than the ammount of stabilizer trim changes.
You can not fly a typical jet transport without intensively using the stabilizer trim, you could probably fly it without using the elevator and just using the stabilizer trim. It is an accepted backup system for the elevator (e.g. on A320, A330/340 in case of total FBW failure)
The elevator is for maneuvres, the stabilizer trim is for all long term pitch changes.
There have been several instances where the stabilizer ended up at the stops and the elevator lacked authority to maintain attitude.
Yes, sthe stabilizer trim is much more powerful than the elevator, hence you may end at the end stops of the elevator before you have the desired pitch change if the stabilizer is trimmed wrongly for the situation you are in.
It is a somehow inevitable for most modern aircraft flying over a wide speed range, having a sophisticated high lift system, powerful engines mounted above or below C/G and being optimized for low drag. You would need a much larger stabilizer and more powerful elevator hydraulics to prevent this situation, or an all flying stabilizer like on the L1011.
Once again unfortunately there is a lot of different longitundinal trim systems (especially in smaller aircraft, but also for example on the BAe 146 which does
not have a stabilizer trim), which prevents pilots to be trained consistently when learning to fly. Hence there is no consistent policy how to use trim, it is type specific which does not help. Also the amount and type of autotrim is differing significantly between aircraft types, as already mentioned for the STS, which actually is the opposite of "normal" autotrim und just usd as an "automatic in manual flight".
I seriously doubt that all pilots have a full understanding of the longitudinal trim system of their aircraft.