The case of the disappearing angle units, or "the dangle of the angle"

[Geoff]

A radian is not an independent unit as you are using it above. It is a dimensionless name.

In the ASTM SI-10 document a radian in base units is m/m (from Table 4), which self cancels out.

Quoting from clause 3.4.4.1 of SI-10 (2016) :

The coherent SI unit of plane angle is the number one; thus, the coherent SI units of the quantities angle, angular velocity, and angular acceleration are, respectively, 1, 1/s, and 1/s^2 . However, it is often convenient to use the special name radian (rad) instead of the number 1 when expressing the values of these quantities.

Note: ASTM SI-10 is titled the American National Standard for Metric Practice.