Sure you can. I can (or could back in the day). The problem is that its pretty hard, and/or requires a very "on-top-of-it" DM. Certainly those were rare, if the constant complaining about bad DMs is any indication. Personally, I did it when I was in college and had plenty of mental time/effort to put into such things. I could rattle off all sorts of numbers regarding ACs and obscures rules and ruling. Nowadays, I'm quite confident that such mental overhead isn't worth my effort.
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I didn't just read those rulebooks, I studied them. Once you had a deep enough knowledge of the various bits and pieces of the system, a sort of gestalt understanding could arise, and then you could use that to crank out things quickly.
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Secondly, what I called friendly DMing.
This reminds me a lot of my approach to GMing Rolemaster, which was my university and post-university game for nearly 20 years.
I couldn't go back to it, though - your mental overhead comment is spot on in that respect, and also there are too many mechanical bits that get in the way.
I agree. Its given me pretty high standards for what I call a "Narrative game" anymore.
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The funny thing is, much like how a sit-com can evolve into a drama, my campaigns ended up achieving a very high reputation amongst my college crowd.
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Drama is very hard to do, even in games that develop that history as part of character creation. You need a very specific type of player who's willing to invest in the tension(s) and characters quickly. I haven't met many people like that.
Its much more effective to start with a simpler, sillier, game and work up to something that "matters." People invest much more readily and deeply into a character whose story they have experienced "firsthand." Characters need to start with plenty of blank spaces that get filled in later. This is yet another factor in my desire for fast-playing rules. The faster we can build up history, the faster folks will be genuinely invested in their characters.
I suspect that you would find my game too slow and insufficiently narrativist for your taste.
I definitely come to RPGing from the traditional side of things, and use the indie ideas/techniques to strengthen a game that is rooted in traditional fantasy RPGing. I would think of my game as quite traditional except for the pushback I've experienced on these boards when describing my approach to GMing (with respect to metagame mechanics, GM authority over plot, "player entitlement", etc).
When my game achieves "drama", I think it would often be better characterised as melodrama or soap opera. When compared to novels or film, I find RPGing to be a pretty poor medium for conveying subtleties of character or situation, because there is so little deliberate creative control over the way those things emerge in the course of play - multiple creative participants, no scope for rewriting/editing, etc.
The short session that I GMed on the weekend began with the PCs barely alive after their fight with the hydra and its elemental bodyguards, and in two separate groups on the battlefield, separated by steaming water, hot semi-solid rock and lava. As the players debated how to proceed, I described the duergar theurge dropping down on a rope from the cleft in the cavern roof.
The players, and some of the PCs, already knew she was there - she had earlier dropped down a potion to help the invoker PC, and the invoker had signalled her that "the debt would be repayed". After a brief exchange of words between the duergar and a couple of the PCs, she walked past them and picked up the fragment of the Sceptre of Law (= Rod of Seven Parts) that had been cut from the neck of the hydra. The duergar explained that this was her due. The PCs disagreed - they would repay the debt, but a fragment of the Sceptre went beyond any dues that they owed, and she could not have it.
This disagreement ended up being resolved peacefully (mechanically, it was a complexity 2 skill challenge). The fragment of the Sceptre wanted to merge with the other 3 pieces wielded by the invoker PC, and it was clear that the duergar was having trouble controlling it - she was sweating (despite her Resist Fire 10) and paying less and less heed to what the PCs were saying. The invoker defused the situation by suggesting that the two parts of the rod be allowed to fuse, and then it be allowed to choose who should wield it. Each through their piece to the ground - she her one fragment, he his three conjoined fragments - and they fused together. The invoker then picked up the rod without trouble. He then invited her to take her turn. She struggled with it, and then relinquished. The PCs then set off with her to her clan stronghold, to rest and to learn what they can do to repay their debt in some suitable fashion.
As well as that dramatic centrepiece of the session, there was plenty of colour. Some of it was basically comedic - for example, the chaos drow trying to extract a guarantee of safe conduct from the ultra-lawful, devil-worshipping duergar; and the religious and historical discussions between the tiefling paladin and the duergar theurge, with the PC warpriest of Moradin occasionally chiming in from the sidelines. Some other colour was "pipe-laying", to borrow a phrase I learned from Robin Laws Hamlet's hit points - the PCs got to see the duergar orc and ogre slaves, and also got a demonstration of the duergar's Expansion ability in the context of a demonstration of how the duergar maintain control over their ogre servitors. This also created an opportunity for various PCs to evince their own hostility to orcs and ogres (especially the invoker, who is near-genocidal towards the "evil" humanoids, after they sacked and destroyed his home city). The whole episode also allowed the players to consolidate their conception of the place of duergar, and devils, in the cosmological conflict that is the centrepiece of the game. (And the tiefling in particular, in the initial interaction with the duergar theurge, emphasised multiple times that in earlier dealings with the duergar the PCs had always found them honourable and true to their word.)
But the negotiation with the duergar in the hydra cavern was also interspersed with a whole lot of procedural play - working out how the PCs make their way across the punishing terrain, for exampe, and how many hit points this cost them - which the system requires, given that a combat with the duergar was a clear possibility, but which I think you would regard as a significant impediment to resolving the central dramatic conflict.
This is where the "heaviness" of the D&D mechanics, even in 4e, puts a bit of a brake on the "now" in "Story Now".
I'm not sure I like players "confidently taking risks", it sounds a little too much like "astutely calculating odds" to me. How risky is it, if you're confident?
The confidence I had in mind wasn't "confience that I'll win", but rather "confidence that I won't be hosed if I lose". My TPK example upthread is the sort of thing I have in mind - because 4e makes it (mechanically) easy to resolve 0 hp as unconsciousness rather than death, it creates scope for PCs to lose combat without necessarily dying. And because of the way it sets up the cosmology of the Shadowfell, it permits PCs to die without death necessarily being permanent - there are a lot of cosmological players (the Raven Queen, but not just her) who might send a spirit back for some reason, which allows PC death to be treated as a complication rather than a "game over" event.
When players know that losing a conflict doesn't have to mean losing the game, I think they become more willing to take risks with their PCs, and to push things a bit harder, rather than turtling up. Turtling is certainly one enemy of free-flowing, dramatic play, in my experience at least.
I too have high hopes for bounded accuracy. I don't think we've seen enough of the rest to know. 
(Just as I had finally succeeded in attracting the attention of an order of Paladins and creating the hero who should've gone on to defeat me, more's the pity.)
) I keep running into games (usually Indie, anymore) that look great, but somehow don't work out as well in practice. I have high hopes for the Cortex+ System from the Marvel rpg. I'm hoping for a solid fantasy version to bubble up. 
