This is something I can't quite wrap my head around. Why would I want the symbol for a unit to be similar to a constant that I will very often use in the same calculations? That has a good chance of ruining my dimensional analysis if I finish up a calculation later. A bathroom break could convert 2 * pi radians into 1 cycle * radian, or 1 cycle into a unitless 2 * pi.

And these conversions would be correct, wouldn't they? Radians are length ratios, and as such dimensionless anyway.

While not technically required, it's pretty convenient to treat radians as their own "angle" dimension. This lets you distinguish an angular velocity from a simple periodic repetition. The latter happen to rarely (never?) occur naturally, but I run into artificial ones often enough.