Why can perturbed wavefunctions be written as combinations of unperturbed eigenfunctions?

[Bob_for_short]

...Why is it always that is says a small time-dependent perturbation?

Actually the external force or field can be strong, not small. The time-dependent solution exists and can still be represented as a superposition of old basis vectors with time-dependent coefficients.

In case of small perturbation these exact coefficients are well approximated with the perturbation series (a la Taylor series). The perturbation theory technique is developing these coefficients in Taylor series in powers of perturbation. Thus smallness of V(t) guarantees good accuracy of a truncated series (practical convergence).