This sounds more like convergent evolution rather than a shared root. Tread derives from Middle English treden; the difference here is to step vs to roll. The

This sounds more like convergent evolution rather than a shared root. Tread derives from Middle English treden; the difference here is to step vs to roll. The latter definition you cite is a newer usage (from the 20th century) which is more directly analogous to the tread on the bottom of one's shoe. The use of a wheel in a treadmill seems more like an implementation detail; the obvious tie-in (IMO) is "persons treading on steps".