"Well, as we have established, the grip of the tire on the road is the ultimate determining factor, so you want wider and or stickier tires. Preferably, both"
Stickier tires yes indeed, but as far as stopping distance goes width is not a factor AT ALL (goes for surface area in a braking system as well), unless you were talking fade, which you clearly were not.
With regards to stopping distance and friction force there are two different coefficients of friction, static and dynamic. Static friction is when there is no slippage between the two friction surfaces (tire and road in this case). Dynamic friction is when there is slippage between surfaces (i.e., lockup situation). With any two friction surfaces being the same, the coefficient of static friction is always greater than the coefficient of dynamic friction. That is why ABS is so good. It keeps the road and tire in a state of static friction, rather than dynamic.
Think of putting a piece of wood (or anything else for that matter) on a flat surface and slowly raising one end of it to create an angle. You will be able to get that angle pretty steep before the wood begins to slide. Once it starts to slide you will have to decrease the angle dramatically to get the wood to stop moving. That is the static/dynamic coefficients of friction at work in a physical application that anyone can test themselves. It is always the case no matter what two friction surfaces you are talking about. Always.
The equation for friction force is: (friction force)=(coefficient of friction) times (normal "vertical" force)
That is it. Notice there is no variable for surface area anywhere in that equation because the surface area does not matter. Sticky tires just increase that coefficient of static/dynamic friction. Ever wonder how a locomotive can pull such huge loads on knife edges that are the steel wheels and tracks? The coefficients of friction are poor with steel on steel, but the normal force is massive.
Other than this slight oversight, your OP is spot on regarding everything else. I am happy to be able to clear this up and add to a wonderful discussion.
Old egg-headed engineer that loves him some good physics talk