Powering out of stall MIGHT be adequate on large propeller aircraft - eg Electra and, I would imagine, the Herc would be similar. Certainly, it was fun to do just that on the Electra - a few shudders and away it would go. However, the size of the prop disks meant a sudden and dramatic change in airflow over the wings. It's a bit different with a jet.
The general certification stall recovery is a bit different to the training version - another example of operations not reading the certification stuff. Some aircraft, if held into the stall will do strange things not addressed during certification. The trick is to get the wing unstalled first (and promptly) ... then worry about thrust.
Indeed with a high alpha a heavy handed bunch of thrust might well produce an unwanted and significant nose up pitching moment rather than the nose down pitching moment desired.
Of course, if this is all happening at near dot feet, then one might be advised to incline to minimising height loss on the basis that you are going to die if the stall persists - a case of playing the risk numbers. That is a specific case with priorities different to the general.