Hmm, I wonder though, do you have any references showing prior understanding by the physics community at large that the r in the Schw. Sol. is the Gaussian Curvature of radius (assuming you agree with that determination, but it is easy to check)?
As I understand the definition of Gaussian curvature, the Gaussian curvature of a Euclidean 2-sphere of radius r is 1/(r^2).
I am not a physicist, and can't speak for the physics community at large. It's easy to show that, for at least the past 40 years, the relativists have been aware of this relationship between Gaussian curvature and the Schwarzschild radial coordinate r. That relationship is implied by statements such as
Hawking & Ellis:
The coordinate r in this metric form is intrinsically defined by the requirement that 4πr2 is the area of these surfaces of transitivity.
In my previous post, I cited the equations given in MTW and in Wald, which make that relationship even more explicit than in Hawking&Ellis. I also quoted Wald's lucid explanation of the important distinction that must be made between the Schwarzschild radial coordinate r and the more intuitive notion of the distance between a 2-sphere and its center.