the nut does play a part in tuning stability because it's a point of contact + it has allowances for movements (vibrations more like it) or the lack of it.
if you have a vibrato bridge, it's best you get a locking nut, otherwise, your machine heads/ tuners should be locking.
for fixed bridge & vibrato equipped guitars alike, the nut slots have to be incised well + lubricated. every time strings vibrate (especially so when open strings are picked), if they aren't stretched out properly, there would be movements at the nut (however minimal).
if the nut slots are choked, this would prevent necessary movements there when your bend your strings hence tuning anomalies would entail.