Anticipating possible future deployment of connected and automated vehicles (CAVs), cooperative autonomous driving at intersections has been studied by many works in control theory and intelligent transportation across decades. Simultaneously, recent parallel works in robotics have devised efficient algorithms for multi-agent path finding (MAPF), though often in environments with simplified kinematics. In this work, we hybridize insights and algorithms from MAPF with the structure and heuristics of optimizing the crossing order of CAVs at signal-free intersections. We devise an optimal and complete algorithm, Order-based Search with Kinematics Arrival Time Scheduling (OBS-KATS), which significantly outperforms existing algorithms, fixed heuristics, and prioritized planning with KATS. The performance is maintained under different vehicle arrival rates, lane lengths, crossing speeds, and control horizon. Through ablations and dissections, we offer insight on the contributing factors to OBS-KATS's performance. Our work is directly applicable to many similarly scaled traffic and multi-robot scenarios with directed lanes.