I’m fairly confident that frictional coefficient is dependent on area, among other things. Don’t confuse frictional coefficient with frictional force.
the coeff. of static friction is independant, of area. YET, the total static friction IS a function of area as well as the coeff.
So…
Friction Force = (Normal Force) x (Coefficient of Friction)
You’re stating that as area increases frictional force increases (which I agree with) while coefficient of friction remains the same. I disagree with this because that would mean that normal force would be the increasing variable.
If you have a greater contact patch size we both agree that Friction Force increases, so the other side of the equation increases as well. Normal force (i.e. the car’s weight) remains the same, so coefficient of friction must increase.