When part of the tire gets flattened on the road surface, then saying there is a radius of the tire (or wheel) doesn't really make sense since it is not a circle. That's why the term "effective radius" is used. What really matters is the perimeter of the outside of the tire (which is not circular) as it rolls across the road surface. Effective radius is this non-circular perimeter divided by 2pi. It is the radius of an imaginary circular tire that has the same perimeter as the actual non-circular tire perimeter.
When you push a tire against the road surface the arc length of the flattened surface of the tire is shorter than when it was unflattened. But in order to shorten the arc length there has to be a force acting in the direction of the arc length. The only such force is from friction. If there is no friction then there is no force and the flattened section simply spreads out to maintain its original length. If there is no friction then effective radius = original radius. If the friction is high enough that it causes a "stick" condition (once a point on the tire touches the ground it doesn't move or slip) then the effective radius turns out to be the distance from the center of the wheel to the ground. If the friction is somewhere in between, then the effective radius will be somewhere in between.
I assume with your next to last sentence you mean that higher volume, low pressure tires change their cal constant more than low volume, high pressure tires due to temperature changes. Generally I agree, but the difference may be due to other tire characteristics that are almost always associated with higher volume tires, such as thicker rubber and wider ground contact area.