They could just make truckers pay it all. They do the vast majority of damage to the roads as it is. For side roads we could just pay the old fashioned way with normal taxes. People without cars still benefit from the roads existence.
The question is how to move the funds around, as state taxes and road funding stop at state borders, but vehicles don't. And of course states will have a patchwork of registration rules, and won't share money with each other. So the trucking industry will lobby some state to be a tax haven, and all trucks will just be registered there.
If only we had some sort of organization that allows states to solve issues that happen between states, like a larger government function...
I’ve seen numbers that state the road damage caused by one truck is equivalent to several ten thousand cars. (Usually numbers between 10k and 40k).
That seemed hard to believe when I first heard it, but the fourth power you mention is the explanation for that insane number.
On one hand they get a bunch of kickbacks for their reelection war-chest and some roads fail 5 years after their re-election is over anyway and someone else will be blamed for it. On the other hand they directly fund their opponent who is unlikely to turn down those dollars, possibly fail in their reelection bid, and successfully save some roads that everyone will just assume would have lasted anyway.
It's not hard to see that it's practically impossible for any other outcome, the system is practically designed to create this kind of situation.