I only have two suggestion. Even though you put in a new key, it could be marked now and that is all it takes OR the blade is slipping and not carrying on thru acting as an extra source of inertia energy. This would be typical of some B&S push mower engines. You didn't say what it is.

Walt Conner

Walt: As soon as it warms up a bit, i'm going to look at it again. I'll take your suggestions into my mind. I hate to take it all apart again, but i will if i find nothing out of kilter. Thanks: Rusty J.
PS: The key looked ok, and not nudged, but i'll check it again. And they are always good for parts. good wheels, and carb and other stuff! THKS!

here are a couple more ideas.

Failed mechanical compression release
Excessive valve clearance
liquid lock

some of the Briggs coils have electronic advance. I don't know if a failed coil could be operating in the advanced mode all the time.

Since this is a "freebie" my guess is that it is an older one, no advance for instance.

Rusty, first thing I would do is look at that blade. Those older B&S had a very light flywheel. IF the blade had a "slip" mechanism and slipped just a little when hitting compression, that was enough to make them kick back.

Walt Conner

Well, i'm going to drag it back in, and check it again! And, if it doesn't go right, it'll go left to the scrap yard! Thanks for the advice! RJ

Walt: the more i think about it--the blade looked like it had been replaced. Maybe it is loose, but not slippy loose, or not the correct blade! Ah-hah! Something to check after lunch! Thanks: Rusty

Well, i changed the blade, but, it still does it! It got pushed outside, to moulder with the other junkers, waiting for their ride to the scrap yard! But, if i can find another engine, i'll use the deck for it! Thanks for the suggestions, though.
Rusty J.

A refresher on the mower engine: A friend suggested that i install two gaskets between the large square plastic filter/ primer bulb/filter holder plate where it bolts onto the carb inlet area, so i did that. It fixed it!! Don't know why or how, but it is docile as a lamb when i pull the starter rope, and runs well, too. Any ideas why? RJ. And, i installed a different blade and holder on the crank-shaft, but that did nothing for it!

With one gasket, sometimes the air filter housing doesn't seal tight against the carb and when you prime, you don't get enough fuel into the throttle body. Most people just prime a few extra times. This is because the plastic housing slightly warps over time. Sometimes the air leak is so much that it won't prime at all.

With two gaskets, you make up the air gap caused by a warped housing.

Even though it starts, I still don't see the relationship between kickback and not enough primed fuel.

A valve seat which has popped out would cause quick a kick-back. Removal of the head would show this. Maybe give that a whirl.