I move them through positions as soon as they can mentally cope with it. I didn't used to teach by position because I thought I was fundamentally opposed to it, but there were definitely problems with that. There are problems with everything, though, in that there is not a single concept that teaches all concepts at once, and the act of piano playing needs to be broken down into parts in order for students to learn how they function together. And, if an element is not truly learned, then their overall understanding suffers.
Generally, I teach by position, always starting with middle C. Then C Major, and playing HT will begin in these positions, as well as the introduction of flats and sharps in here. Playing in these positions may also involve small crosses, like one finger reaching a note outside of the position. Students generally like having positions to give them a home base, and when they have a home base I can help them learn broader concepts step by step without overwhelming them. I place heavier emphasis on fingering than on note names, and deal with note names and key names "separately" (but thoroughly, and they are of course ultimately connected).
My main reason for this is because I feel that, as far as the hand itself and technique is concerned, it is most important for them to start to understand the principles of fingering as it relates to note groupings in the text, the width and flexibility of their hand, and how to feel the entire hand on the piano at once. I want for them to carry over into any repertoire how the hand relates to groups of notes, and eventually how to make fingering and technical decisions especially when it uses expanded hand and harmonic positions.
With somebody who is an averagely interested learner, and who is practicing somewhat regularly at home, they will learn the concepts in my second paragraph in the first couple of months, and be expanding soon after. Most of these people are seeking the expansion, and as long as somebody is grasping the concepts I want for them to be grasping, and especially if they are seeking to expand, I will not hold them back.