On reading Steve Giddings’s article, I felt the need to share some information about the original work of Karl Schwarzschild.1 

His solution contained no coordinate singularity; it had only the singularity at the origin. That was neither a mistake nor an oversight on Schwarzschild’s part. By choosing a nontrivial scaling factor for the angular part of the metric, he had carefully avoided producing the coordinate singularity. As a side effect, such a choice results in a nonzero surface area for a sphere of zero radius. However, that result should not be a problem since one is not dealing with flat space anyway.

Sitzungsber. K. Preuss. Akad. Wiss.
Jan.–June 1916
, http://www.biodiversitylibrary.org/item/93032#page/31/mode/1up.