To answer your questions in short:
- Why does friction stop exerting a force?
Static friction still works. The wheel is not slipping; it is rolling. At the contact point the wheel and surface stay together because of static friction. This point of the wheel doesn't slide over the surface. So no kinetic friction, but certainly static friction. If not, then how would you start your car? Your tires need grip (static friction) on the surface, and must not slip.
The quote you give does though mention constant velocity. That is the same as saying no acceleration and of course also no angular acceleration. When that is the case, which it will be after some time, all torques must sum up to zero. So, if friction is the only force that causes a torque, then friction must be zero. Friction is only making the wheel start speeding up it's rotation - it makes the wheel start to turn, when your car speeds up. But when the rotation is constant, there is no more friction - just like when the block is sliding with constant speed on the ice, or when the space shuttle is drifting at constant speed, there is no friction that brakes it.
- Am I mistaken that friction exerts a force whenever an object and its surface might be distanced from each other?
Yes, you are mistaken. If I lift a bag of potatoes from the floor, there is no friction.
But I know what you mean: what if you move the object in parallel along the surface. Still, ideally you can have a no-friction icy surface, but apart from that I guess not.