Quasiperiodic systems in one dimension can host nonergodic states, e.g., states localized in position or momentum. Periodic quenches within localized phases yield Floquet eigenstates of the same nature, i.e., spatially localized or ballistic. However, periodic quenches across these two nonergodic phases were thought to produce ergodic diffusivelike states even for noninteracting particles. We show that this expectation is not met at the thermodynamic limit where the system always attains a nonergodic state. We find that ergodicity may be recovered by scaling the Floquet quenching period with system size and determine the corresponding scaling function. Our results suggest that, while the fraction of spatially localized or ballistic states depends on the model's details, all Floquet eigenstates belong to one of these nonergodic categories. Our findings demonstrate that quasiperiodicity hinders ergodicity and thermalization, even in driven systems where these phenomena are commonly expected.