sbharris, You say The cross product is tricky, but it produces forces in two directions, and only one of them is tangential. 

Not if the v term is already restricted from the derivation to only being a radial term. This is further confirmed by the fact that the Coriolis term in the Cartesian/polar conversion equation is unequivocally in the tangential direction. The maths in this derivation is identical in principle.