I’d rather not spend time on this if anyone knows the answer.
L5 corticothalamic cells definitely project to the brainstem and such, but that doesn’t mean they’re motor outputs. Many corticofugal projections seem to target sensory cells, modifying sensory input rather than generating behavior.
S1 seems to activate a muscle (source), so motor output still seems universal. I’m just wondering whether motor command cells are corticothalamic.