I suspect you're underestimating the effects of the wargaming roots, both on D&D specifically and on role-playing games in general, which, in those early days, were all but synonymous.
The cover of the original edition of D&D, published in 1974, described it as "Rules for Fantastic Medieval Wargames". Although it included various non-wargaming elements, and sometimes took pains to emphasize them, the background this emphasis should be viewed against is that of traditional wargaming, which is what its designers and target audience would've been familiar with. After all, before D&D took off, there was no separate role-playing game community or tradition to draw on — early D&D, like Chainmail, was designed by wargamers, and for wargamers.
Both Chainmail and early D&D grew from wargames through an essentially evolutionary process: the rules, as published, are basically tidied-up snapshots of the house rules that Gygax and Arneson developed for their gaming groups, as they gradually mixed in new elements (such as magic, individual heroes and non-combat challenges) into what was still, at its roots, a wargame, albeit a somewhat oddball one.
The heavy emphasis that early editions of D&D sometimes placed on non-combat elements should thus be seen as an attempt to get the players, assumed to be familiar with traditional wargames, to include any non-combat elements in their games at all. Of course, as the rules were still rather combat-centric, the general effectiveness of these measures is debatable, but at least they did encourage others to continue the exploration of non-combat elements in "fantasy wargames" — now increasingly called "role-playing games", to differentiate the emerging new genre and community from traditional wargames — that Gygax and Arneson had started.
Much of this evolution took the form of other game systems, published in the wake of D&D's success, and often diverging further from D&D's wargaming roots. Thus, D&D itself came to be regarded as a "traditionalist" game, a term often regarded as synonymous with "combat-heavy", while other games experimented with alternative concepts like simplified, less crunchy combat systems, or even eliminating combat entirely.
(For D&D specifically, this reputation was certainly not reduced by the fact that, among the various iterations of D&D reinventing itself, the fourth edition, in particular, took a major step back towards its combat-heavy wargaming roots, perhaps in part to differentiate it from the many competing systems at the time that tried to de-emphasize combat. However, regardless of edition, D&D has always been a rather crunchy and combat-focused system, and that's arguably part of its core identity — take away the combat mechanics, or replace them with something completely different, and many would say that what's left would no longer be D&D in anything but name only.)
So, to sum up, the reason D&D is so focused on fighting is that, originally, it's a wargame with fantasy and role-playing elements mixed in, not a fantasy story-telling game with combat elements mixed in. Furthermore, the history of D&D is inextricably linked with the history of RPGs as a concept, such that, in many ways, D&D can be seen as a "living fossil" of what the whole genre of "role-playing games" originally started from. If D&D today seems combat-heavy, it's only because we're looking at it from a modern perspective, and comparing it to the many other game systems which took the ideas pioneered by D&D and ran with them much faster and further than D&D itself ever did.
Best Answer
The Vorpal swords originate from "Jabberwocky"
Lewis Carroll's children's book "Through the Looking Glass", contains a nonsense poem called "Jabberwocky", which is full of arcane and unusual words. Alice finds the poem in a book written in a seemingly unintelligible language. Realizing that she is traveling through an inverted world, she recognizes that the verses on the pages are written in mirror-writing. She holds a mirror to one of the poems and reads the reflected verse of "Jabberwocky".
In the poem, a boy takes a vorpal sword and slays the Jabberwock, a horrible monster:
The poem depicts the vorpal sword as a weapon that can slice "through and through" and apparently decapitate the monster. It actually does not say it does that. The boy could have cut off the head of the dead monster after he killed it, to bring back as a trophy.
Although it is not explicitly said that the sword chopped off the head, the line about the boy returning with the dead monster's head follows right after slicing through, so it is easy to make that connection, and apparently Gary Gygax did.
There is no other historic usage of the word vorpal that could further explain what it means. It looks like Carroll just invented the word. Martin Gardner, in "The Annotated Alice" reports that Carroll said he could not explain this word (while he explained various other words in the poem). Interestingly, the Collin's Dictionary, defines vorpal to just mean "fatal", and cites its first measurable appearance in literature as 1872, the year after "Through the Looking Glass" was published.
The first Vorpal Sword in Gaming starred in Greyhawk
Gygax turned Lewis Carroll's books into two special levels1 of his original Castle Greyhawk home game that was the testing ground for the D&D rules he published. He included the Vorpal Blade in random loot tables as the result if you rolled a 00 on a d100, and published it in Supplement 1: Greyhawk (1975), where he defined it like this:
Clearly, he read the poem as the sword having a penchant for chopping off heads.
1 Coincidentally these are the only levels of that famous dungeon that were ever published by TSR, EX1 Dungeonland, and EX2 The Land Behind the Magic Mirror -- the latter one contains a vorpal blade (p. 19).