The dynamics of a cornering wheel Can a rolling wheel create a side force without first rotating on a vertical axis?
 A: We tend to ignore the (relative) rotation of the wheel to the car for several reasons.  The rotation itself doesn't cause the forces that are under analysis, so examining them isn't always useful.
I don't exactly know what your question is, but I'll give you two things to think about that might help you be more comfortable with the scenario.
The first is that you might want to think about an (ideal) wheel as a device that has zero coefficient of friction in one direction and infinite coefficient of friction in the perpendicular direction.  So when a force is applied that has components in both directions, you get acceleration (motion) and you get a reaction force.  Finally, if a torque is applied to turn the wheel, it can turn as well.
So this supplies the answer to your main question:

Can a rolling wheel create a side force without first rotating on a vertical axis?

Yes.  If you apply a side force to a wheel, it will supply a counter side force up to the friction limit.  This is independent of the rotation or direction of the wheel.
The other thing is to imagine rather than a real car where (some) wheels can rotate, a toy car where the wheels are fixed.  We'll mount the wheels similar to a car turning.  Now there will never be any (relative) rotation of the wheels to the car.
When the car sits at rest, there are no (horizontal) forces or torques.  Then without changing the axis of the wheels, we start to push the car forward.  The angle that the wheels have causes forces that push the front of the car left, and it causes torques that spin the entire vehicle (and with it, the wheels) counter-clockwise. 

a wheel only rolls in one direction along a straight path.

That's only true if no (vertical) torques are applied to the wheel.  As soon as a torque is applied, the wheel will turn, which could modify the path that it may travel.

I am describing the ideal wheel and how a force ONLY occurs when the rolling wheel rotates/pivots on a vertical axis.

For an ideal wheel, there are no forces required to pivot the wheel.  A real rubber tire has a contact patch and friction which detracts from the ideal, but other solutions are possible. 
Turning forces come about when a force is applied to the wheel that is not in line with the movement direction of the wheel.   

We currently believe that when we attach the wheel to a vehicle with other wheel(s) pointing a different direction it travels a circular path by way of a different principle.

The reason I gave the example of a car where the pivot angle is fixed is to show that you can go from no turning happening (car is stationary) to turning just by applying a force forward on the car.  You don't have to change the pivot angle.  
You have asked twice about seeing a problem, and I must say that I don't think I understand your specific concern.  I'm probably missing the critical point.
