First off, as a story, the tale of the Alliance destroying Orgrimmar isn't narratively satisfying. The Alliance has long been established as the less brutal, more 'noble' faction. They're the guys who don't blow up cities. Neither Horde nor Alliance players are going to be happy in the long run with a story where the Alliance becomes brutal and warmongering without significant lead time, and absolutely no Horde player (which is half the game, remember) wants to play the story of how the Alliance blew up their capital city. Look back to the reaction of Alliance players over the loss of places like Theramore, Southshore, or Azshara (which wasn't really even an Alliance territory in game) and imagine how much worse the loss of the major capital city would be. Furthermore, it's not a compelling story idea to base the next expansion around, and you couldn't well ignore it. Something as drastic as the bulldozing of Orgrimmar would demand that its story be told in full. Do you want that expansion?
Perhaps you do. Clearly, some players do, or are at least willing to argue for it. But before we really discuss the narrative, let's look at what Horde players have already had to absorb over the past two expansions. If you came into the Horde with World of Warcraft, you experienced a group that felt entirely different from the Horde of the Warcraft RTS games, especially I and II - you were presented with a Horde that had reasons for its actions, that presented the player with an experience of being oppressed, united by adversity. The reason so many Horde players use terms like 'underdogs' and 'banded together for mutual survival' is entirely because that's how the narrative they experienced framed them - it actually did a brilliant job of fostering faction antagonism by giving both sides realistic grievances and issues with the other.
It would be my argument, however, that World of Warcraft actually works better when the narrative doesn't center around Horde/Alliance conflict. That conflict should be there - high tensions between the two factions are part and parcel of the setting - but the increasing levels in Cataclysm and Mists have placed the two groups front and center in a way that ends up creating too much antagonism, and too many accusations of bias. The reason the destruction of Orgrimmar would be a nightmare is because it would force us to have another expansion centered directly around Horde/Alliance conflict. While I'd argue Siege of Orgrimmar is the best raid of an expansion with several good raids in it, I at least am beyond ready to get back to fighting someone else for a while.
Of course there's always the standard answers as to why we're not going to go beyond a certain point - deposing Hellscream - and seeing further change. First off, Cataclysm proved that some change is too much change - again I point to the outcry from Alliance players over losing Southshore, a tiny spec of a town which required a great deal of effort for players to even reach it in the old days. Completely alienating half the player base by wiping out their major city isn't something that seems particularly wise to me, especially not now, after an expansion where many of those players had their very conception of the Horde challenged. You see a lot of Horde players saying "I feel like the story is on rails" because they had no real choice until patch 5.3 - they couldn't really say no to the quests telling them to do questionable things. Heck, those players who were fine with Garrosh's actions couldn't even opt out of the Darkspear Revolution - you could do none of the quests, if you wanted to avoid it, but that just left you doing nothing. Forcing them to participate in a major raid (remember, Horde players are raiding Siege as well) that ends in the destruction of their own city would probably be too much for the narrative to bear. It wouldn't be a compelling story, nor would it be particularly sweeping and heroic for either faction.
Ultimately, the truth is we need Orgrimmar right where it is when the dust settles. The complete destruction of the city would rob the setting of the faction tension - it's possible to imagine an entire expansion built around the bedraggled survivors of Orgrimmar finding refuge and the Horde rebuilding from such a major defeat, but that would be an expansion totally about the Horde. What would the Alliance do in this hypothetical expansion? Be happy? Would your Alliance characters be going to victory parades and welcomed by cheering throngs while the Horde players had to slog through swamps and grasslands with a band of refugees? The Horde characters would at least have stuff to do in that scenario. The Alliance would be sitting around being glad that they got rid of the Horde, at least for a while. Probably it would then fall apart without a clear sense of an enemy to keep them all unified.
All of this is fairly meta-argument, of course - it does not address why Varian didn't push things more when he was face to face with Vol'jin. That's because Varian already explained his reasoning, and it makes sense to me. Remember, the Alliance couldn't take Orgrimmar. It needed the help of pretty much every Horde faction leader except Garrosh, and a huge chunk of the Horde's divided military force - trolls, tauren, blood elves, forsaken all fought against Garrosh to some extent. To even attempt to conquer Orgrimmar fully now would mean fighting against all of those forces. Could such a war be won? Possibly. Trying to push it now could cause the rebellious forces to join forces with what remains of Garrosh's loyalists to push you out, and even if you won that, you're then faced with two choices. Either you commit genocide to keep the Horde population in line, or you attempt to absorb the population of the Horde territories into your own. Does the Alliance stand strong enough yet to have imperial ambitions? Can it occupy and govern all, or even a fraction, of the Horde's territories with hostile Horde populations working against them at every step? Ultimately, for the Alliance, what's the end game of all this?
Yes, he could have made some demands. Doing so puts him in a position where he has to be prepared for either eventuality - either Vol'jin capitulates to his demands (looking weak immediately to his own people, making the likelihood of his own fall and replacement with an unknown more probably) or he refuses, and then Varian either has to back down (looking weak to his own people) or he has to be willing to wage total war, on the heels of the battle they just fought, in order to gain what he just demanded - gains he admits later he doesn't even think the Alliance can properly garrison and hold. The Horde spent three years constantly preparing for war. The Alliance did not. In order to win the day, the Alliance would have to be prepared to not only kill a great deal of Horde, but it would have to be willing to sacrifice a huge number of its own people. Why do it, when there's a chance of getting out of Orgrimmar with a clear victory, a strong position, and the lives of most of your people intact, as well as the possibility of a Warchief you at least understand?
It neither suits the Alliance from a narrative perspective, nor from the perspective of its players, to make this story any more about Orgrimmar or the Horde than it already has been. It's time to move on. It's time to escape from Orgrimmar.