I've been reading about GR recently and I can follow the derivation of a Schwarzschild solution to it's final and well known form in Schwarzschild coordinates.
The orbit stability argument (for a massive test particle) is also clear - no stable circular orbit can exists for $r<6M$.
What usually follows after that is a calculation for the Earth:
$r = 6GM/c^2 = 0.03m$
radius of the Earth $= 6300km$.
So comparing them one notes that it is not a problem for the Earth because 0.03m is well below the surface.
My question is - how can we make such a comparision? Radius of a planet is measured in spherical coordinates but $r$ in $r=6M$ is in Schwartzschild coordinates - while deriving Schwartschild solution one starts with spherical coordinates but makes a lot of coordinates transformations so the resulting $r$ is really a very complicated function of a spherical radius and comparing their values seems wrong.