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How does area affect friction *after* the tires are skidding?
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The equation is the same but the coefficient is different (lower). That is why it is important not to enter a skid, since when that happens you immediately lower the available grip, hence ABS. Contact patch has the same (ir)relevance either skidding or rolling (except when the surface under you can move).
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as far as contact patch goes, wider is better in the sense that more traction becomes available be each square each has to work LESS to hold the car in place. no matter what type tire you are using, a wider version of that same compound/tread design will loose traction later.
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Each square each has to work LESS to hold the car in place
but has less ability to do so because it has less reaction force. Its like replacing 8 horses to pull a cart with 16 Shetland ponies so they can do less work each. If they can only pull half as much then they put out the same pulling rate. A given compound on a given surface has a coefficient of static friction, the available friction (force) increases linearly according to that coefficient and the reaction force between the surfaces. Area is not factored into the equation.
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the real issue becomes balancing ALL of these needs into a vehicle that can take daily driving abuse in climates from the equator to the poles, all at reasonable pricing.
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You only need to balance the needs that you actually have not the needs that some other human on a different part of the world may have. I highly doubt anyone regularly makes trips from the equator to the arctic. Each person will have their own needs depending on their driving environment. I live in the city/suburbs of Sydney and never have to drive in mud or snow so it would be a waste to buy all terrain tyres. That's why there are speciality tires which you can cater to your needs. I think the real issue is finding the MOST EFFICIENT tyre that can still meets your needs, which will be less for people in snowy/unpaved regions but the principle still holds. An ecomodder might even decide it is worth an afternoons work and higher initial cost to swap a winter and summer set of tyres so they aren't meeting needs which aren't there.