How did Bradley arrive at the +/- correct speed of light when he calculated it?
He didn't need to know the absolute position of the stars. The change of the appearent position  has been sufficient to get a good estimate.
For velocites small in comparison to the speed of light, the shift of the angle is still small.

For small angles the sine is proportional to the angle with 2nd order precision.
The first Taylor summands are
$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - + ... $$

The 2nd order approximation $$\sin x\approx x$$ can be applied on stars perpendicular, or at least apearently perpendicular, to the plane of the Earth orbit, where the method is most accurate.

Hence the underlying reason, for which it works, is the smoothness of sine.

This argument can be adjusted to less optimal conditions.