the police do not have to wait until a gun is pointed at them before they fire.
Obviously. In fact the person does not even have to be holding a gun at all. He could be holding a taco for all they care, but if the cop wrongly mistakes that taco for a gun he feels he has free reign to open fire. And therein lies the problem.
By the way, the Tueller drill is pseudo-scientific nonsense based on a misinterpreted article from a trade magazine article in the 1980's. It was originally posed as a piece of "rule of thumb" style advice in that article and has not been scientifically verified in a controlled test.
Absolute nonsense. Thousands of civilian, military, and law enforcement academies have run these type of drills and show the danger of someone with an edged weapon despite some distance between the suspect and offender. The behavior that you are demonstrating now, is typical of "experts" who often pop up on internet message boards. The anonymity and safety of the internet lets lots of people without a clue pretend to be an "expert" in everything from law to police and military tactics, despite the fact that they have no training, education, or experience in any of the above. It's an amazing thing to watch.