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".
(The word is not use much nowadays; archaic, as the dictionary entry says.)
I was only speculating about the connection between the two words (trundle and treadmill), because they sound vaguely similar. Maybe they have a common Latin or Old English or Old German root, as many English words do.