Hey Narundomestics,
You guys are turning me into a physics instructor! I will try and answer but what exactly you want to know is not very clear in your questions.
It does not mater what diameter, width, or style tire is on a car, “rubber on the road or contact patch" will remain identical providing the weight of the car and the air pressure in the tire are the same. Archimedes Law stated that a floating body would immerse itself in a liquid until it displaces a volume of water equal to its own weight. There is a corollary here, the tire contact patches will continue to enlarge as the vehicle sinks down, until the contact patches are large enough so the internal pressure against the patches equals the weight of the vehicle. This is because of the air pressure in the tire pressing down against the patch, is the equal and opposite reaction of the air pressure at the top of the tire, Newton’s Law is holding up the car.
A vehicle will sink down until the force pushing up in the top of the tire, which is no longer balanced by the pressure in the bottom of the tire, because the pavement is taking the pressure through the tread, is equal to the vehicle weight. It makes no difference if the tires are wide, narrow, large in diameter or small, high or low profile, the tire contact patch size, in square inches will be the same for any given vehicle weight and tire pressure.
Why wide tires then?
You might recall a demonstration in physics class where a brick is pulled along a board attached to a string and a fish scale. It made no difference if the brick was on its wide side, narrow side or standing on end, the amount of force to pull the brick was the same. The explanation given was the coefficient of dynamic friction of the board and the brick remains the same. The brick remained the same weight, on end more weight per square inch, on its side less weight per square inch but more square inches. As the weight per square inch varied proportionately, there was no change in friction so long as the surfaces were constant.
Yes, this seems to contrdict the effects of wider tires?
There are some other considerations that play into it with tires though.
Tire compound, usually performance tires are softer and a softer tire will allow the rubber to conform to the road better, providing a higher coefficient of dynamic friction. .

Tire “A "on the left has 18 square inches of contact patch , tire “B” on the right has 18 square inches of contact patch.
It would take an awful lot of writing to discribe all the variables, so I will try giving some quick examples of other factors which demonstrate why a perfomance difference between a wide and a narrow tire.
Tire “A” hits a 4-inch wide hole, its adhesion or traction will be momentarily lost, while tire “B” still has 4 inches of contact patch left on the road, and more friction available to move the car foward.
Tire “A” hits a 4-inch bump it will be completely affected while only half of tire “B” will and it is more likely to absorb the bump then bounce off it, more friction.
Tire “A” in an emergency stop or under hard acceleration has the same rubber in contact with the road twice as long as tire “B”, resulting in a greater amount of heated rubber compound and less friction.
A narrow tire must have high sidewalls in order to have a large contact patch or the rim will have insufficient clearance above the road. High sidewalls generate more sidewall flex and cornering distortion, more heat less friction.
Tire “B” has 18 square inches of contact patch with far less side wall flex, thus less heat build up and more friction.
I hope this answers most of your questions if not, let me know and I will try and explain it in another way.