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My understanding of an aspect of it:
The theory that a wider tyre won't give more grip is because basic mechanics of smooth body friction (what we are taught in schools) doesn't use contact area as part of the equation. F=mu*r, where F is the force due to friction, mu is the coefficient of friction between the two surfaces and r is the load normal to the contact patch. On this basis it would be impossible to turn, accelerate or brake at more than 1g.
But real life isn't "Smooth Bodies" (except, arguably, on catwalks). The tyres are not merely using tangential friction, but digging into the surface of the road and pushing directly against it. There is also a chemical bonding function that nobody ever goes into any detail about and just assures us it happens (if anyone knows more about the actual chemical bonds that occur I'd love to know more in detail).
As such, more area DOES give more grip.
The shape of the contact patch defines the directions in which the grip can be generated. A wide, small diameter tyre will have a wide, thin contact patch, and will provide grip in a different way to the same area that is narrow and thick (if you see what I mean), although I think it's unfair to suggest that wide&thin = lateral and narrow&thick = longitudinal. But F1 cars use wide tyres when not regulated away from them, whilst dragsters use narrower taller tyres to get good straight line performance. Obviously drag, wear, heat management and so forth will play their parts in the choice too.
But essentially, and ignoring variables that could make the opposite true if you wanted it to, like silly camber angles, wider tyres give more grip.
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