I joined looking for an answer to this question. Did a search, unsuccesfully, but spent a lot of time reading various discussions. Great forum.
A long time ago (1975, maybe) I had a Peugeot 10 speed. I could ride no hands forever on a smooth road, around curves and even corners. No bike I have ridden recently has been so stable. Last week I dragged my in-law's old Raleight 3 speed out of the basement and got it running. I took it for a ride, found it very comfortable and stable, and found I could still ride no hands (not that I would anymore).
I'm guessing the difference is something in the frame geometry, but just what, and why? More important, do they still make bikes that will ride like that? I think I would like such a bike for general riding around on the roads and bike paths.