Do CFGs generate many languages?
Every grammar, by definition, generates a single, unique language.

The definition of "generating $L$" includes that $\overline{L}$ has to be not generated. That's obviously not the case for any proper subset of the "largest" generated language.

The definition has not been chosen without reason: using your approach, every formal language would be regular (as subset of $\Sigma^*$). That would not be a useful model at all.