Once the person eventually reaches enough seniority in the system to warrant joining the management team as a check captain, there is a mysterious absence of psychological testing for the job. Yet these people are responsible for flight standards and may even hold the office of a civil aviation Delegate.
First of all, training is not management. It is training. Of course the postholder training and the fleet training managers are management position, normal trainers are not. Besides, who says there is no psychological assessment? Or stuff like classroom presentations of training courses before a panel of senior traning staff? All done in the companies i worked for, the assessment is not held inhouse but by external psychological asssessors who specialize in aviation and indeed work for the same company that does entry pilot selection for our cadets or direct entry pilots.
Yes, some of the old style trainers are still there, and they used to be the disciplinary arm of the management in the olden days, but those times are long gone and the last ones will retire soon. Since nobody can get checked out of the job anymore they have a surprisingly changed attitude nowadays too.