Great example with Hinckley, Jay. And, as you said, adrenaline definitely throws off aim and concentration, and can also interfere with decision-making. It's also true that, while the saying ridicules bringing a knife to a gun fight, that understates the deadliness of a knife-wielder. When my relative went through the academy they showed the class a demonstration of a man with a knife about 10-15 feet from a trainee with his handgun holstered. The 'attacker' moved with deliberation toward the trainee, striding, not running. In most cases the trainee didn't even get their gun fully out, let alone get off a good shot, before they were 'stabbed'. And if a person is speaking at the same time, as is often the case during police-involved shootings, that slows reactions down more.
One thought I had is that I'm surprised there aren't more instances where there are head shots reported, at least for those who are shot while approaching the officer. My relative was trained to put 2 in center of mass, and if the person kept going, to put the 3rd shot in center of forehead. I don't know how many forces use similar training, but the idea was, as Peter B said, to make sure the person was stopped, and that last shot (especially with a .357 magnum) is pretty much guaranteed to do that.