How can you write an algorithm to properly fill a circle using lines from the center?

You have chosen wrong method to find all touched cells - instead of point-based way you need cell(squares)-based approach - ray intersects rectangle rather than point.

There is article of Amanatides and Woo "A Fast Voxel Traversal Algorithm for Ray Tracing" for 2D.

Practical implementation.

Example:

Quick-made tracing example. Rays emitted from left top corner go to blue points. If ray meets black cell obstacle, it stops. Pink cells are lighted by rays, grey ones are not.