I'll try to answer by giving elements that are, I suspect, responsible to a certain extent about that. But I doubt there is an official text about why choices were made toward what we have now, so all in all, my answer is quite opinion-based.
First steps from wargame
As you said D&D was at first some kind of spin-off from Chainmail, think like Mordheim for Warhammer Battle if you are not old enough. Even if you have common fantasy literature inspiration, the wargame influence is stronger than you assume in your question. Hell, it is "Dungeon & Dragons, Rules for Fantastic Medieval Wargames Campaigns Playable with Paper and Pencil and Miniature Figures", it even assumed that the players got their own chainmail figurines and knew its rules (basic information on the wikipedia page). At this point, D&D was, in my opinion, more like a support rule to adapt your wargame to smaller scale, not a standalone product and no universe was described in the basic rules (important for later).
Wizards of the Coast
Now, it could have evolved toward more roleplay focused rules like some RPG did. I think that this path was not taken because of WOTC. Let's have a close look to their products, we can see that they publish the books for the most popular universe for D&D. Actually, for D&D3.5, the three core books didn't provide universe descriptions, but rather advertised more books to describe these universe, with their own change in rules. They also still relied on using figurines, which WOTC can also provide. So, here again, great wargame component.
Let's go back on the books, they became the main source of inspiration for how D&D evolved, and books such as Dragonlance or those in the forgotten realm have a heavy combat component (more than less RPG-tied books), they evolved along D&D and exerted a mutual influence. D&D is now totally influenced by WOTC-edited books rather than any other fantasy book, in fact, they are part of the same 'product'. EDIT: it did not started with WOTC buying TSR, it in fact started before, but WOTC bought it all, probably as part of the D&D product.
I think that WOTC intended D&D to become some sort of common RPG resource to play a lot of product they would edit, d20 system tend to confirm this right? So, they provided what people needed the most as core, fighting rules. Roleplay and social interaction being too much universe-tied to fit in this format.
At that point, another challenger appeared: video games. Games such as Baldur's Gate, or later Neverwinter Nights became popular and used the 3rd version of D&D, they featured way more fights than a classical pen&paper campaign. In 2006, we even had D&D online. Then D&D4 was released in 2008, and it took inspiration of all these video games back from where it came. A lot of people indeed said that D&D 4 was clearly a video game on paper, I'm not sure about the true goal of WOTC, but it was clearly designed to provide a simpler set of rules, and a lot of tricks for everyone to have fun in fights (warrior's special powers and so on), so that young gamers who never heard about pen&paper RPG could easily access it. Video games, just like wargames before, heavily influenced the evolution of D&D.
So that's it, I mainly think that D&D became more and more combat oriented because of commercial intents. In my opinion, it's also easier to sell combat-oriented add-on, look at how much they published for D&D 3 and 4, compare the number of lore rulebooks (Forgotten Realms Campaign Guide) to power gaming (guide of the complete ultimate warrior/rogue/bard/whatever... of doom). Power gaming has this thing, linked to the need to get more material, that compel players to buy stuff, whereas more narrative/roleplay players tend to come up with their own home-made stuff. You could compare to this very Stack Exchange, a lot of question are fight related, or system related, fewer are purely roleplay related, I think the reason is simply that when you do narrative roleplay, you find your own answers as a group rather than seeking the arbitration of an official rulebook.