We propose a human-centered safety filter (HCSF) for shared autonomy that significantly enhances system safety without compromising human agency. Our HCSF is built on a neural safety value function, which we first learn scalably through black-box interactions and then use at deployment to enforce a novel state-action control barrier function (Q-CBF) safety constraint. Since this Q-CBF safety filter does not require any knowledge of the system dynamics for both synthesis and runtime safety monitoring and intervention, our method applies readily to complex, black-box shared autonomy systems. Notably, our HCSF's CBF-based interventions modify the human's actions minimally and smoothly, avoiding the abrupt, last-moment corrections delivered by many conventional safety filters. We validate our approach in a comprehensive in-person user study using Assetto Corsa-a high-fidelity car racing simulator with black-box dynamics-to assess robustness in "driving on the edge" scenarios. We compare both trajectory data and drivers' perceptions of our HCSF assistance against unassisted driving and a conventional safety filter. Experimental results show that 1) compared to having no assistance, our HCSF improves both safety and user satisfaction without compromising human agency or comfort, and 2) relative to a conventional safety filter, our proposed HCSF boosts human agency, comfort, and satisfaction while maintaining robustness.