(I recall reading that the damage is proportional to the vehicle weight to the fourth power, probably with some more nuance)

Yes. Road damage is based on vehicle weight. To the 4th power, yes. Heavier vehicles do exponentially more damage than lighter ones. https://www.hagerty.co.uk/articles/opinion/opinion-cars-have-a-weight-problem-and-its-damaging-more-than-the-environment/

But actually it's based on axle weight. This is why Semis have many axles, to spread the weight out.

But actually it's based on tire weight. This is why Semis have doubled wheels on their axles.

But actually it's based on contact pressure. This is why Semis have wider tires than your standard car.