A post by Fighting Fantasist got me thinking about this.
Variable Weapon Damage is more realistic and more importantly players understand it better as it reflect everyday experience. D&D combat may be abstract but it still grounded in reality. A target needs to be hit, that it will take damage after being hit.
If Hit Points measure the life expectancy of a character "under fire" then it will be reduced faster with hits by a two handed sword than a dagger. Whatever speed advantage smaller weapons have is far outweighed by the mechanical advantage and mass of larger weapons. For this reason many accepted variable weapon damage as a natural extensions of D&D.
Simpler and abstract combat systems appeal to many. And should be accepted as such rather than trying to hang a fig leaf on them. I seen too many over the years wind up complex combat rules because of referee's fig leaf. Because somebody said "You know Y is true because of X." From my view the d6 damage system of White Box only D&D doesn't need to justified. The elements of White Box D&D combine to produce a fun game with a specific feel that many enjoy to this day.
There are many who proposed a additional rules that work with the d6 system without sacrificing it. One of my favorites is where you roll 2d6 damage and take the highest to represent two handed swords and polearms. Likewise with dagger and other small weapons you could have a rule where you take 2d6 and take the lowest.