The problem with Diablo 3 was never that the AH made trading official, or too efficient. The issue has always been that the loot drop system is so bad, making the AH a virtual requirement for playing the game at all. But loot dropping is supposedly being improved in RoS (though I hope we're not moving too far to the other extreme of the spectrum, the console's flood-of-legendaries).
In addition, it's clear that Blizzard is making many more items BoA in RoS. From what I understand, items modified by the Mystic will be BoA. Demonic Essences are BoA, and I expect several other important crafting and enchanting ingredients to become BoA as well. Crafted items are BoA. If much of the best gear and much of the ingredients required to make the best gear is BoA, it can't be sold anyway, even if we kept the AH.
These changes will (hopefully) significantly reduce our reliance on the AH, to the point that removing it is not really necessary, and will not have much effect on the game as a whole.
However, removing the AH will have an effect on the part of the game that it's meant to influence: trading. I for one rather like that trading is easy and safe nowadays. (Safe except for the 0 gold auction bug, anyway...) I'm not looking forward to once again joining random WTS or WTB games in hopes of finding someone who isn't asking too much for their items, or offering too little for mine.
You only need to look at the AH today to see how many people are absolutely clueless about the true value of items. People selling items for 10 times their true worth is the rule, not the exception. Forums are marginally better than trading games, but searching through them is still much harder than it needs to be, and comparing different items to each other is all but impossible.
On the bright side, a lot has changed over the past 13 years, since the release of Diablo 2, both in gaming and on the internet. We are not stuck with trade games and forums. I see no reason why someone could not implement the Diablo 3 AH as a third party website, allowing people to put items up for sale, search for items and filter them by affixes and cost. It wouldn't be quite as convenient as the current ingame AH, but it could do the job. Indeed, such sites already exist, and I imagine their userbase will grow explosively when RoS hits.
Ultimately, the removal of the AH will have less impact on the game than people think, because the presence of the AH wasn't what caused the problem in the first place. The AH just exacerbated a different problem: the crappiness of the item and drop systems. The fact (hope) that Blizzard are fixing those problems already makes us less reliant on the AH. Removing the AH on top of those changes is neither necessary nor particularly important. In short: http://www.youtube.com/watch?v=cDGlN6mluGA