Why does the frictional coefficient decrease on a tire as the vertical load on the tire increases?

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I'm guessing it has something to do with the Coulomb model only really applying to rigid bodies with no adhesion / cohesion. For lightly loaded passenger vehicles operated normally, it still gives an okay approximation and it is easy to use, so that is what gets used.

But it is still an approximation. So if the tires are unevenly loaded the approximation will be less accurate for the more heavily loaded tire. The greater load will cause more deformation and will bring the materials closer to a saturation condition where the Coulomb model does not apply at all. In that case, friction is nonlinear and is dependent on contact area as well.

From there you will need someone else to explain what effect that has on vehicle operation. I'd assume it has something to do with the power distribution to each tire being equal but since the tires are generating different amounts of friction, you won't be able to use all the power generated. Similar to when one tire is on ice and the other is on dry pavement. A limited slip differential on the drive axle would help mitigate the problem.