Theory predicts the superconductor-to-insulator transition (SIT) to emerge from the competition between Anderson localization, which tends to localize single-particle wavefunctions, and superconductivity, which establishes long-range correlations in the superconducting order parameter. In two-dimensional (2D) superconducting films, the transition temperature Tc at which resistance vanishes, R_Box(TBKT)=0, is set by the Berezinskii-Kosterlitz-Thouless (BKT) mechanism and satisfies TBKT< T_c0, where T_c0 is the mean-field transition temperature. In weakly disordered samples TBKT≲ T_c0, whereas increasing disorder drives TBKT≪ T_c0 near the SIT. Whether the finite-temperature transition retains its BKT character throughout this crossover remains an open question. Here, we investigate the evolution of both sheet resistance R_Box(T) and superfluid stiffness Jₛ(T) over a wide range of disorder strength W. We establish that even near the SIT, the finite-temperature transition from the superconducting to the resistive state remains of BKT type. However, as disorder approaches the critical value, the zero temperature superfluid phase stiffness, Jₛ(0), is found to vanish rapidly while T_c0 remains finite, which we attribute to quantum phase fluctuations as the drive for the zero-temperature transition. Three decades after its experimental discovery by Haviland, Liu, and Goldman, our measurements clarify the origin of the SIT in 2D films.