First some geometry/word definitions.
Flat only means a constant plane. That plane can be oriented arbitrarily in space.
Running true means that the plane is oriented perpendicular to the axis of rotation in both remaining axes*(think x,y,z)(right up away).
Now a flat face plate rotating on a shaft will 'wobble' if the shaft is bent.(plane is not perpendicular to the axis of rotation etc.).
A flat workpiece will 'wobble' if it's flat surface is not perpendicular to the axis of rotation etc.).
I believe you may be experiencing wobble, not machine flexing.
Good advice re gouge above. Also videos.
*I looked up the plural of axis!!