Besides the interesting discussion about radians considered or not as dimensionless units (just read about why they shouldn't be dimensionless [1]), you can have your own simple QuantityMagnitude that effectively normalizes radians:

quantityMagnitudeRadians[q_, unit_] := QuantityMagnitude[q/Quantity[1, unit]]

This would work for all these cases:

In[64]:= quantityMagnitudeRadians[5, "Radians"]
Out[64]= 5

In[59]:= quantityMagnitudeRadians[Quantity[4, "Radians"], "Radians"]
Out[59]= 4

In[60]:= quantityMagnitudeRadians[Quantity[4, 1/"Seconds"], "Radians"/"Seconds"]
Out[60]= 4

In[61]:= quantityMagnitudeRadians[Quantity[4, "AngularDegrees"], "Radians"]
Out[61]= \[Pi]/45

In[62]:= quantityMagnitudeRadians[Quantity[4, 1/"Seconds"], "Radians"/"Seconds"]
Out[62]= 4

In[63]:= quantityMagnitudeRadians[Quantity[4, "AngularDegrees"/"Seconds"], "Radians"/"Seconds"]
Out[63]= \[Pi]/45

[1] https://iopscience.iop.org/article/10.1088/1681-7575/ab27d7

Thanks for the quick response. I probably should have given a better example. This works:

QuantityMagnitude[Quantity[1, "AngularDegrees"/"Seconds"], 
 "Radians"/"Seconds"]

but this doesn't:

QuantityMagnitude[Quantity[1, 1/"Seconds"], "Radians"/"Seconds"]

Either way, the point is that you should be able to convert a dimensionless quantity to a quantity with units of Radians, because Radians are unitless (they are defined as the ratio of two lengths)