I would argue that dimensionful quantities are never used as arguments to power series, however you are correct that this does not imply that angles are dimensionless, since (like we do with other quantities, we can divide by whatever unit we like to get something dimensionless). I withdraw that argument.
A better argument that angles are dimensionless is that dimensionless quantities are formed by the ratio of two quantities with the same dimension, and the angle subtended by an arc in a circle is given by the ratio of the arc length and the radius.
That argument holds just as true for dimensional quantities frequently computed by power series though, which means it can't be valid.