You're correct that the entropy should strictly be considered. However, the Second Law doesn't say that entropy is always maximized. At constant temperature and pressure, instead, it's the Gibbs free energy $G=U−TS+PV$ that's minimized. The three terms on the right indicate that it's thermodynamically preferred to (1) be in a low-energy state, (2) with a large number of macrostates (which is connected to the Boltzmann entropy $S$), (3) without doing much work to expand into the external atmosphere. As you note, the third term isn't important for this discussion.
That leaves us with a tradeoff between potential energy and entropy that depends on the temperature. In fact, free energy minimization drives some charge carriers to move inside the sphere, because the entropy increase from the first few carriers exploring the inside of the sphere is relatively large. The potential energy term, however, ensures that the vast majority of carriers remain on the surface. An increase in temperature would make the entropy term more important and would drive more carriers into the interior despite the energy penalty of them being closer.
(The same argument applies to vacancies inside the polycrystalline structure of a metal conductor and surface steps on its surface. These are also equilibrium structures that spontaneously arise precisely from the entropy consideration you're asking about—even through they cost energy in the form of broken atomic bonds—and they also arise as a strong function of temperature. In each case, the microscopic interpretation is that the fluctuating energies that give rise to the bulk definition of temperature allow particles to do new and unusual things.)
Thus, Griffiths is simplifying the situation somewhat by ignoring the entropy contribution because it's assumed to be negligible in this case (relative to the electrostatic benefit of having charge carriers be widely spaced). He's considering the low-temperature limit.
