Trouble with the lids of cylinders I'm having trouble making, and meshing cylindical-like objects. In both cases below I drew some vertices in the XZ plane and used screw to rotate. Except screw did not seem to like me having objects on the axis of rotation, so I deleted those nodes and filled. In the example on the left, I have filled with a cylinder. I then applied a metallic texture. Cool. Almost perfect, except you can see that remnant on the top that was rotated by screw has left radial sector marks.

So then I deleted the cylinder and deleted all of the faces along the top, and bottom, and replaced with GridFill. The mesh looked not so bad at all. But the texture [on right] is not better. Maybe worse. I tried adding a seam, which you can see here:

The unwrap did not look like what I was expecting. I thought I'd get two circles and a mesh, but all I get is a mesh
Grrr....
If I understand your description, without the bevels, your model would look like this in x-ray mode:

If I select all and unwrap using the U key and selecting 'unwrap', I get this:

I think you might want to mark two more circles for unwrapping so the ledge area unwraps cleanly?

unwraps to: