We present a hierarchical safe auto-taxiing framework to enhance the automated ground operations of multiple unmanned aircraft systems (multi-UAS). The auto-taxiing problem becomes particularly challenging due to (i) unknown disturbances, such as crosswind affecting the aircraft dynamics, (ii) taxiway incursions due to unplanned obstacles, and (iii) spatiotemporal conflicts at the intersections between multiple entry points in the taxiway. To address these issues, we propose a hierarchical framework, i.e., SAFE-TAXI, combining centralized spatiotemporal planning with decentralized MPC-CBF-based control to safely navigate the aircraft through the taxiway while avoiding intersection conflicts and unplanned obstacles (e.g., other aircraft or ground vehicles). Our proposed framework decouples the auto-taxiing problem temporally into conflict resolution and motion planning, respectively. Conflict resolution is handled in a centralized manner by computing conflict-aware reference trajectories for each aircraft. In contrast, safety assurance from unplanned obstacles is handled by an MPC-CBF-based controller implemented in a decentralized manner. We demonstrate the effectiveness of our proposed framework through numerical simulations and experimentally validate it using Night Vapor, a small-scale fixed-wing test platform.