The ideal tax would probably be weight + miles, but good luck implementing it...
A really naive scheme[1] would be scales at the pump that affect gas price. Weight-class based pricing would also work. It's possible to enforce with the surveillance state apparatus and microparticle-tagged fuel products.
1. Without taking into account the incentives nor human nature
Your intuition is correct. It's one of the few times we get to use the fourth power so it is aptly named The Fourth Power Law[1].
Of course, higher taxes on the heaviest vehicles would just mean higher prices on consumer goods, so this mostly ends up as a subsidy to consumption.
I would also argue that if you want to make a comparison that you just find the, I am sure, studies that have been conducted by every department of transportation in the country to estimate the fair cost-of-use to vehicle type. The comment here should either find some references or be removed as snark-bait.
"Sure, in the sense that O(n) is the same as O(n^4)"...
But road wear simply is not proportional to gas use. If you double a vehicle's weight, its gas usage will almost certainly less than double (this depends somewhat on driving patterns, but air resistance is generally a significant factor in mileage), while the road wear it causes will increase by roughly a factor of 8. A flat tax on gas can't possibly internalize the costs of road wear, though it could internalize the costs of CO2 emissions, which are actually proportional to use.
The problem becomes convincing anyone there's political credit to be gained by implementing/overhauling those systems. I suspect it will take as-long-as-it-possibly-can for those and bureaucratic reasons (contracts/pilots/implementation of those systems).