I hope this adds something here. From my types you will see I haven't been hands on for a while.
What I was taught in engineering training is that the pusher in the 737 was called a stick nudger and is purely there to properly identify the stall by pushing the column forward gently. The stick shaker similarly is there to replicate the pre stall buffet for those aircraft whose characteristics need additional identification.
(Now showing my age)
The Trident
and the BAC 1-11 had true stick pushers and were a recovery system. Being a T tail configuration they couldn't be allowed to stall fully as they wouldn't recover after the tail enters turbulent air from the wing as some poor souls found out in testing.
In those aircraft the stick pusher operates rapidly pushing the control columns forward while the tail is still in clean air. Pitch control is this case was taken away from your good selves as it wasnt possible to restrain the columns. In the case of the 1-11 accompanied by very loud klaxons.