Loop cut maintaining circular profile Adding a loop cut to an object with a circular profile renders the object non-circular.  This is true for any object with a circular frame e.g. plain circles, UV-spheres, cylinders, cones, etc.

Top-left is the mesh of a plain circle with Subsurf modifier applied (bottom-left).  Notice how adding one additional vertex (top-right) makes the circle skewed (bottom-right).
Is there a tool/transform/way to correct the circle?
Knowing the solution to this should help solve the problem for all the aforementioned objects.  I tried selecting all points and applied the To Sphere transform in vain.  It didn't help.
I thought putting the vertex r radius away from the circle's centre like the rest of the vertices will help, looks like it doesn't.
Since the verts are non-uniformly spaced, the result will always be deformed.
That's just how the Catmull-Clark subdivision works.
You can use Cast Modifier to correct the circular profile.
It can also be limited to a vertex group.

Add Cast Modifier &gt; Sphere
Set Factor = 1
Done