Briefly, the generator of $x$-translations is more precisely the $x$-derivative $\frac{\partial}{\partial x}=i\frac{\hat{p}}{\hbar}=i\hat{k}$, the wavenumber operator.

The generator of $x$-translations is more precisely the $x$-derivative $$\frac{\partial}{\partial x}~=~i\frac{\hat{p}}{\hbar}~=~i\hat{k},$$ the wavenumber operator, which is independent of $\hbar$.