From a modern viewpoint, the calculation leading to the result GM/R^2 is
merely an attempt to determine the magnitude of a tidal effect, that is,
the *difference* between the acceleration of freely falling bodies at the
surface of a spherically symmetric distribution of mass M and radius R and
that of freely falling bodies at its center. (For instance, a similar
calculation will show that the tidal effect between either pole and any
position on the equator has a magnitude of 2^(1/2) GM/R^2.) It is only if
you can assume
1) that "the" acceleration of the center of the sphere is zero (whatever
that means!),
2) that there is no centrifugal or other inertial contributions to the
tidal effect, and
3) that there are no other contributions to the tidal effect from nearby
mass distributions (like the Moon or the Sun)
that the quantity GM/R^2 can be equated with "the" acceleration of freely
falling bodies at the surface.