"With the power those things have, simply lowering the nose to the horizon should see them fly out of a stall in a few seconds."

For an initial stall, quite right. But deep stalls result in a flight path so steep the stalling angle is significantly exceeded due flight path alone.. As most transport category jets have underslung engines, applying thrust results in a pitch up moment, making the problem worse, not better. It's for this reason that Airbus have removed the use of thrust from their published stall recovery technique.

It's so critical that post stall (ie.when you're back to a flying AoA), overzealous thrust application can result in a return to stalled flight. Can be a bit of a giggle to watch newbies in the sim learn this the hard way.

CJ