Trucks pay higher tax rates for diesel, but I don't think the point of use fees must be to charge people for how much damage they do to the roads but how much benefit they obtain from the roads.
This is very tricky because roads have so many positive externalities. Even if you live in a walkable neighborhood and don't own a car, you will walk to stores that have goods delivered to them by truck, right to your neighborhood, so you can walk up and buy them. Without all those trucks bringing things close to where you live, you wouldn't have that convenience and would need to get a car yourself. Probably a heavy car since you'd have to haul everything yourself -- everyone would need to drive right up to the factory gate and then transport it back for every good you buy. Imagine what that would do to the road quality! So trucks help use the road more efficiently and actually save the road relative the alternative of not having dedicated trucking.
And not only that, but that fact means local stores are in competition with stores farther away, which drives down prices. So even people who never drive obtain many valuable benefits from the roads that connect their neighborhood to other neighborhoods, and many of these benefits are delivered, so to speak, by trucks. Thus everyone pays for these benefits.
This idea that once there is a road connecting two markets, that prices begin to equalize in them was one of the things observed by Adam Smith when he talked about how local shopkeepers often opposed roads connecting their town to towns with larger markets (and cheaper goods). Now they don't equalize perfectly, they equalize up to transportation costs. Thus lowering transportation costs for the trucks that ship goods between markets results in lower prices for everyone who buys those goods, even if they don't use the road themselves. This is why it makes sense to subsidize the transportation of goods.
Thus our current system makes sense, roads are paid in part from general funds, in part from use fees, and there are some extra fees added to trucks.