You're correct that contact patch area is primarily a function of pressure and weight. But the formula above is an approximation under the assumption that all parts of the tire in the contact patch area press down equally hard on the road surface. With real tires that's not the case and there's more pressure in the middle of the patch than at the edges.

But I disagree on your contention about cornering ability. Car and motorcycle races depend critically on cornering ability and if there were an inherent benefit to skinny tires then you'd find them in those sports. But the opposite is true - with quite wide tires being used. OTOH, in bicycle racing the most important concern is minimizing drag and weight since the available power is so limited. A skinny tire wins here in three ways: 1)air resistance is less, 2) higher pressures can be used for a given weight and construction therefore reducing rolling resistance, and 3) the weight will be less. And yes, the wider tire actually has a little less rolling resistance *if* both are run at the same pressure and are otherwise similar. But in practice the skinny tire can be run at much higher pressure than a wide tire of similar construction and at that higher pressure the rolling resistance will be less.

But for commuters these effects are pretty small. I regularly switch between my road bike with 23mm tires and my touring bike which currently has 38mm and 35mm tires. Can't say that I really notice much effect on my speed even in fairly fast group rides. But I'm pretty sure they'd become significant in any racing situation.