The Kruskal-Szekeres "Extension": Counter-Examples

The Kruskal-Szekeres "coordinates: are said to "extend" the so-called "Schwarzschild solution", to remove an alleged "coordinate singularity" at the event horizon of a black hole at $r = 2m$, leaving an infinitely dense point-mass singularity at "the origin" $r = 0$. However, the assumption that the point at the centre of spherical symmetry of the "Schwarzschild solution" is at "the origin" $r = 0$ is erroneous, and so the Kruskal-Szekeres "extension" is invalid; demonstrated herein by simple counter-examples.