I don't think that's true in most situations. I think most companies would still pay a severance out of a general effort to smooth things over, and some sense of social loyalties.

And some aspects of the agreement would be fine. No spilling actual business relevant secrets, nothing that would count as libel or slander, etc.

Nope, they'd have security show you the door and maybe threaten to sue you into the ground if you talk. They choose mkney because money is easier, but if you take this option away, you just get the stick and no carrot.

Part of smoothing things over is implicit of explicit understanding that employee also smooths things over that is shut up after receiving the bribe.

Companies are people, many people have positive feelings towards co-workers and people like them

True but not the ones who will be firing you. At least not in big corp, it would be some HR worker who doesn't know you from Adam and couldn't care less about you.

They always have the manager, and the director and possibly VP will be in the loop and know what going on.

The immediate manager will be informed, but they can do nothing about it or in any way change the process or the result.