According to literature contact patch pressure distribution is not constant.
The probably most simple approximation is parabolic 3 / 4 (1 - x) (1 + x), which results in the following tire friction curve:
Now this is something we can work with.
Playing a bit with the parameters we get this:
The right half of the curve is constant (tire fully sliding) due to the simple constant sliding friction model I am using. Using Stribeck friction function should fix this.
I'd also like to try a few other distributions that should match the real pressure distribution a bit better: a skewed parabolic and a simpler trapezoid function.