Here you aren't cancelling out any units, you're just neglecting to write them down, and scaling things so that outcomes of interest happen to land at 0 and 1. Expecting special insight to come out of that operation is numerology.

Hmm. You are right, and I should fix that. When we did that trick in school, we always called it "dimensionless", but you are right it's distinct from the pi-theorem stuff (reynolds number, etc). I'll rethink it.

Edit: Wait a minute, on closer inspection, your criticism seems to apply to radians (why radius?) and reynolds number (characteristic length and velocity are rather arbitrary in some problems).

Why are some unit systems "dimensionless", and others not? More relevently, taboo "dimensionless", why are radians better (as they clearly are) than degrees or grads or arc-minutes? Why is it useful to pick the obvious characteristic lengths and velocities for Re, as opposed to something else.

For radians, it seems to be something to do with euler's identity and the mathematical foundations of sin and cos, but I don't know how arbitrary those are, off the top of my head.

For Re, I'm pretty sure it's exactly so that you can do numerology by comparing your reynolds number to reynolds numbers in other problems where you used the same charcteristic length (if you used D for your L in both cases, your numerology will work, if not, not).

I think this works the same in my "dimensionless" utility tricks. If we are consistent about it, it lets us do (certain forms of) numerology without hazard.