Variable Weapon Damage

Is Rolling a D6 Good Enough?
In Volume I, Men & Magic a successful attack deals 1-6 points of damage determined by the roll of a single d6. Supplement I, Greyhawk introduces variable weapon damage whereby different weapons cause different amounts of damage as determined by rolling different dice. The Advanced Game (and all subsequent editions) also uses the concept of variable weapon damage and introduces a weapon type "to hit" adjustment table including data  on weapon speed, reach and length and room required as well as a matrix of "to hit" adjustments based on weapon verses armor type. (Some gamers, including the group I mostly play with ignore this table.) Why is this?
In the absence of any specific comment by Gary Gygax, principle author of Supplement I and the Advanced Players Handbook, I venture to guess why this amendment to the combat mechanic may have occurred. Perhaps it was to give the impression the game is more crunchy, more data driven and realistic with the inclusion of variable weapon damage and the "to hit" adjustment table. The Original Game emphasizes speed of play using abstract combat mechanics. Much of the detail of what is happening during the 1 minute combat round is left to the players' imagination. In White Box both hit points and weapon damage are randomly determined by rolling the d6. If hit points represent mostly the character's ability to avoid the killing blow as some old grognards claim then all weapons, being deadly instruments, can chip away at this ability by threatening harm, coming very close to seriously wounding and causing the PC to use up their luck and bag of tricks to avoid serious contact with said weapon.
All editions of the World's Most Popular FRPG rely on abstraction and the use of imagination in their game mechanics. White Box is especially so and it is one of the reasons I enjoy it. Realism in a game is an illusion, perhaps even more so in a game based on the fantastic. Suspending disbelief helps us to become immersed in the fiction and push through the illusion in order to share in some of the adventure experience through the avatar of our character. Some of us find pacing and narrative description more important than crunch in that endeavor.
If damage represents the reduction of one's ability to avoid the telling blow, to avoid character death or a mortal wound, then it makes sense to me that all weapons, natural and man-made, have roughly equal potential for causing this effect. The many variables which remain unspecified, but effect the outcome of the attack, can be represented by the random score of the die roll. If hit points come in units of six, it rather makes sense that the reduction of hit points also be measured in units of 1-6.