I don’t think you need a 3D physics engine. 3D rendering technology came along, on PC and consoles like the PS1, N64, long before there were mature, stable 3D physics engines and libraries. The CPUs were just not up to it, mostly.
Generally early '3D' games used either 2D physics (side-on for platformers, top-down for driving games) or none at all. Certainly something like Mario-Kart, where the focus is on fun not physical simulation, you could get away with a combination of 2D dynamics and special case code.