It's that simple. If it isn't working, I don't know what to tell you. Vert spacing along the loop doesn't matter; as long as it's a proper edge loop it should turn it into a planar circle:
That first screenshot is bad, don't select the edges of those inner polygons. Select the loop you want to become the circle. The inner shape is there because as you've noticed, without it the Circle function creates a mess. Those inner edges don't form a proper edge loop due to the corner verts only being connected to…