I've much more generally characterized the circumstances under which the abbreviated form of division matches the correct quotient. This allowed me to explain why it works in Is this a consecutive-prime/constant-exponent number? and part of Shift right by half a bit. I still need to prove why the algorithm itself has these characteristics though, with adequate rigor.

I feel I am closer to fully explaining the prime power part of it, but not the 3(quotient) > divisor part. It's a bit strange finding a 3 here; maybe has something to do with 4 being the first composite number. But it actually would be 3(quotient) >= divisor part if not for 12/6 failing and matching the quotient 4 instead of 2.