Comment by algas

That first result re: tires is simply wrong. Wider tires don't have a larger contact patch; the size of the contact patch is determined by the weight of the car and the air pressure in the tires:

    A = W / P

So the reason wider tires improve handling is more complex and subtle. Also, FTA:

    Assuming a baseline of a moderately wide tire for the wheel size.
      - Scaling both of these to make both wider than the OEM tire (but still running a setup that fits in the car without serious modifications) generally gives better dry braking and better lap times.
      - In wet conditions, wider setups often have better braking distances (though this depends a lot on the specific setup) and better lap times, but also aquaplane at lower speeds.
      - Just increasing the wheel width and using the same tire generally gives you better lap times, within reason.
      - Just increasing the tire width and leaving wheel width fixed generally results in worse lap times.

A full accounting of the effects of changing tire width should explain all of these effects.