A bicycle actually consists of two inverted pendulums (the frame and the fork) joined by a hinge. And the bike that was stable with no trail and no gyroscopic effects used this to provide stability; the stability was provided because the fork, with the lower center of gravity, would fall faster than the frame (taller inverted pendulums fall slower than shorter ones, which you can easily demonstrate by how much easier it is to balance a long object like a shovel than a short object like a spoon on your hand). This provided the necessary feedback that caused it to steer into a turn in a way that stabilized.

The takeaway is that there are several factors which influence the stability of a bike. We know of certain designs which utilize one or more of these factors to achieve self-stability (and the conventional bike has all of these factors, hence why it tends to work so well), but we don't know the exact set of conditions on the combination of factors which would allow you to characterize which designs are stable versus unstable, without simply trying out any given design and simulating it.