Good explanations. My terminology before in FABO was not quite right. The "shopping cart effect" is actually termed "trail", as the wikipedia article explains. Positive "trail" causes the front wheels to tend to follow the current direction of the car, as in pulling it with a tow-bar.
Caster is the vertical angle of the wheel pivot axis in a side view. I might be wrong (again), but it seems that both caster angle and trail are involved in the tendency for the wheels to self-center. Shopping cart wheels have large positive trail, but zero caster and don't self-center. If there was large positive caster but zero trail, it seems the wheels would also not self-center.
I know that in bicycles (and motorcycles), the steering axis must angle back (positive caster) and hit the road at a point behind the tire contact patch for the steering to self-center, i.e. you can ride with no hands. That is why beach bikes have forks that curve at the bottom and motorcycles have the pivot behind the forks. Many high-tech road bicycles have straight forks and are very unforgiving if you let go of the handle bars. Bikes are thus stable with "negative trail", which is the opposite of what wikipedia suggests is needed to self-center. I suspect the SAI angle also comes into play in cars.
Designing a front suspension is very involved. When the wheels turn, the inner and outer front wheels must angle differently. I get concerned when people on FABO suggest mod'ing a rack and pinon steering from a junkyard car to a Dart. You might get it to fit, but what about all those subtleties, some of which could put you in a ditch.