I disagree with your suggestion to promote the radian to the International System of Units (SI units) to address the confusion over dimensionless numbers (Nature 548, 135; 2017). On the contrary, this could perpetrate the mistaken view that everything in the physical world has dimensions.
The humble planar angle is an example. As the ratio of the length of a swept arc to the radius of that arc, it is dimensionless. Typically, however, people append the radian label directly to the resulting number, or attach other tags — such as degrees, gradians or minutes of arc — having used other conversion factors.
You mention the regrettable practice of referring to torque units in joules per radian. The mathematical distinction between torque and energy is clear: torque is the vector product of the radial 'lever arm' and the applied force. Its units are strictly newton metres. Joules are applied only if this torque moves through a unitless vector rotation, with the scalar product yielding the scalar quantity of energy.
If the radian were to be promoted to an SI unit, a further point of caution is that formal dimensionless analysis would require ad hoc patching because of unbalanced numbers of terms containing the radian label. This is especially the case for established dimensionless groups that have angular frequency components (such as the Strouhal, Rossby and Womersley numbers).
In my view, the clear teaching of principles is more important than unwarranted tinkering with the consistency of the SI system.
Otherwise, we risk throwing the dimensionless baby out with the unit-ambiguous bathwater.