[Christ]

How would the contact behind the kingpin cause it to want to turn further in?

Casters work on this principle already... they always move to center when trailing...

Move the kingpin behind the contact patch of the tire, and it doesn't matter what (reasonable) caster angle you have, the forward momentum is going to cause oversteer as the road friction tries to push the tire further off center, instead of further toward center.

Edited to clarify -

I know that Macpherson strut and any other kinds of steering/suspension use the system that you're talking about, but those systems also use an offset steering angle with positive force on one tire helping the return, due to there being two tires, one always going slightly faster than the other in turns.

Now we've got one tire, trailing a single line on the surface. Don't think it's gonna work out the same way.