That historical and disputed assessment is that damage goes as 4th power of axle load.
In other words, 10% overweight causes 50% more damage, 10% underweight is 35% less damage.
It is a matter of where to draw the line and balance cost of extra traffic - including extra weight and cost of a truck cab, extra driver, and extra pavement to accommodate more vehicles vs still existing damage from weather patterns.
There has been research since the AASHO Road Test that suggests the actual power law relationship can vary to some degree based on various factors such as subgrade type and degree of compaction, pavement type/thickness/service history, extent of usage by overweight vehicles, and so on--but the interval I've read of runs from power of three to power of five.
I think it's important to realize (as you suggest) that, whatever the specifics are for a given combination of subgrade, pavement, and usage pattern, it is important to understand the costs of catering to truck traffic and make appropriate provision, whether that involves revising engineering standards for new construction/full-depth reconstruction or somehow shaping the truck traffic through restrictions (e.g., banning trucks on certain roads, taking into account the degree to which such bans will be observed) or economic regulation (e.g., higher taxes or more stringent licensing for trucking-related activities). It is a multi-parameter problem of optimizing consumer's surplus.
The ATA, being a trucking industry lobbyist, is going to be for trucks über alles, and thus has a vested interest in pointing out the fourth-power law does not hold with exactitude in all cases.