"For example, the technique book says "lower end of 3" (this would be for 1-2-3 natural turn, for example)without saying exactly when this is."
That is lower at the end of STEP three, not the end of BEAT three.
Even if you make the assumption that each step takes exactly one beat, there is absolutely zero evidence to suggest that beats align with steps. In fact, there is a lot of evidence to suggest that steps begin and end somewhere around halfway through each beat - that would put the weight change which is approximately the midway action of each step on the downbeat, and would have us counting steps in terms of beats as and-one, and-two, and-three.