Griffiths's derivation looks pretty correct and straight-forward to me, although I hate it if derivations in a text rely on exercises to be solved by the reader. Of course, as an author this spares you to type a lot of details of a calculation ;-)).
Anyway, what he does is to separate the problem of averaging of the microscopic field over a macroscopically small but microscopically large region in a part "far" (i.e., far on a microscopic scale) from the point in question and one "near" (on a microscopic scale). So he was taking out a sphere of microscopically large but macroscopically small extent of the integral first. The remaining integral can be treated with the macrocopic field for the region outside of the sphere, and the remainder integral can be calculated exactly in terms of the charge distribution.
The more I look at Griffiths electrodynamics when reading and posting in this forum the more I come to the conclusion that Jackson might be more advanced but in the long term saves a lot of trouble, because his writing is much more to the point. For a more basic treatment, I think Vol. II of the Feynman lectures is the best source to learn classical electromagnetics. Another very nice book is the famous vol. 2 of the Berkely physics course, written by Purcell.
My personal favorites are the books by Schwinger on Classical Electrodynamics, Sommerfeld's volume II of his Lectures on Theoretical Physics (despite the use of the ict convention in the part on special relativity), and the classic by Abraham and Becker.
