My unedumicated view is that, if the front wheels stayed vertical, and the rest of the body did not lean, it would have a tendency to roll. However, as the front wheels lean into the turn, any rolling force has to work against that wheel lean. The mass of the vehicle holds it down. David makes comment in his videos about it not rolling. The affect is the same as for racing wheelchairs and trikes that have a fixed lean on the front wheels. As Popshot rightly pointed out, 2-wheelers fall over unless they are moving forward and, the slower you go while turning, the more likely they are to fall over.
I would imagine that the velocipede provided time for people to get used to such a thing so that when pedals were attached, the balance was already there and the guy who invented pedals didn't give up on the first day. Remember when you were learning to ride a bike. I remember avoiding hitting a fence, aided by the fact that there was a bull in the paddock, and running into a steel drum instead. If I was the first bike rider, I might have given up. The next thing they need to get right is the seat (saddle).