Kinetic inductance 𝐿_kin quantifies the low-temperature electrodynamics of superconducting microwave circuitry. We directly probe the inductive response in the broadband reactance of superconducting granular aluminum and titanium nitride from MHz to GHz frequencies, Im(𝑍) = 𝜔 · 𝐿_kin, by broadband Corbino reflectometry. Furthermore, we observe an unconventional collective mode featuring an acoustic dispersion relation for up to 10 harmonics, which enables us to obtain the kinetic inductance independently. We report low-temperature kinetic inductances up to 29 nH/sq for a titanium nitride thin film in the vicinity of the superconductor-insulator transition. We calculate the low-temperature superfluid stiffness 𝐽₀ and find it converging to 𝑇_c, consistent with a phase-driven superconducting transition.