Do wide tires have more cornering grip than narrower tires?

Answer

Your explanation in the ‘context of the question’ above is correct. Indeed, the size of the contact surface has no influence on the frictional force.

Wide tires have no more grip than narrow tires when cornering. The technical reasons for choosing wider tires have to do with issues such as heat dissipation and shear stress. And not with grip on the road.

This is certainly counter-intuitive. The following explanation can help you better understand the phenomenon:
Suppose you suddenly take your tires twice as wide. The contact surface is therefore twice as large. The car still weighs the same, and because the contact area has been doubled, the pressure of the tires on the road will be halved (pressure=force/area). Because the pressure has been halved, the surfaces of the tire and road surface only interact half as well, and the frictional force per unit area is halved. The frictional force per unit area has therefore been halved, but fortunately we also had twice as much surface area. The whole is therefore a null operation. The frictional force on the road will remain the same.