This problem is not really as hard to solve as some believe, except getting the airlines to go along with it and to put up the cash to fund it.
TacomaSailor is dead on correct. As an IT professional for over 30 years, I've participated in automating a number of different physical processes (including some aviation processes), and his description of skills erosion due to automation could not have been stated more succinctly, and this clearly applies to aviation as well.
Pilot's are being trained to understand and respond to the automation system's hardware and software that they manage, but not enough training is being provided for the pilot to understand and correctly operate the "physical-ware" the automation is controlling. In my opinion if you don't understand and can't operate the physical-ware, you don't really understand the automation system controlling it either.
The solution lies is the airlines owning a very small fleet of low cost (to acquire and operate) suitable aircraft, where line pilots can hand fly and practice various hand flying skills such as stall prevention, stall recovery, upset recovery, and high altitude work to name a few. These aircraft could also allow some basic interaction and failure mode training between various on-board automation systems and the physical-ware of the aircraft. This would keep the hand flying skills current, and unexpected systems and physical-ware interactions specific to a type could then be practiced in that type's sim. In other words, the low cost aircraft would focus training mainly on hand flying the physical-ware and some basic automation failure interactions, while a type's sim would focus training mainly on that type's specific system failure modes and aircraft interactions.
Getting the airlines to go along with this kind of solution and funding it is the hard part of this solution.