'Spinning is not caused by a faulty differential'
Not necessarily so. Various internal faults or stripped axle end splines could cause such a problem, though USUALLY accompanied by strange noises. What I would so is support the rear wheels off the ground on jack stands, and engage 'drive' (keep at idle). I would observe whether both wheels spin, or if anything seems unusual if I try to slow one wheel down with a gloved hand. Note the dangers involved, and take appropriate precautions if you do this. Do NOT rev the engine.