The common explanation that you mention hinges on the property that the rolling wheel is free to tilt and turn, affected only by the rolling motion and whatever features the terrain has.
If the wheel is constrained in any way (in this case a mechanical constraint that maintains a particular tilt angle) then the precession effect is prevented from occurring.
The precession effect requires motion. The wheel is in rolling motion, when in addition to that there is a tilting motion the response is a turning motion.
With the depicted setup (caster wheel sustaining a tilt angle) the contraption will respond differently to bumps in the terrain as compared to the case of the free wheel.
I'm not sure whether the caster wheels supported wheel will turn. If it does it will (I guess) be due to camber steering, not due to some precession effect.
(I you search with the terms 'motorcycle camber steering' you will probably find good information on camber steering. Motorcycles have rather fat tires, making the camber steering effect more pronounced.)
When tilted, the fact that the wheel is tilted is in itself not a cause for precession effect. You need a tilting motion. Just a small tilting motion will do, and the subsequent turning will tend to return the wheel to upright.