Exercises 6, 7, and 8 in section 10.4 of Hodges' big model theory textbook contain an outline of a proof of the consistency of the following statement

There exists a countable first-order theory $T$ such that $T$ has no saturated models.

Call this statement $\mathrm{NSat}_\omega$ and call the analogous statement with no cardinality restriction $\mathrm{NSat}$.

The proof outlined by Hodges uses a result of Woodin's that it is consistent with $\mathrm{ZFC}$ that for every infinite cardinal $\kappa$, $2^\kappa=\kappa^{++}$, but he mentions that this result assumes some large cardinals and in particular 'a supercompact is more than enough'.

Are large cardinals necessary for this result? There are different meanings of a question like this but a typical formulation would be something like:

Does $\mathrm{ZFC}$ + 'there exists a transitive model of $\mathrm{ZFC}+\mathrm{NSat}_\omega$' prove the existence of transitive models of any typical large cardinal axiom? What about $\mathrm{ZFC}+\mathrm{NSat}$?