We could imbues bitcoin with intrinsic value if we use bitcoin as a notarization database.
The problem with that is the use of bitcoin as a notarization service will erode bitcoin as a form of money. You would need to pay higher fee to prioritize bitcoin over big documents such as DNS record and contracts. That become inconvenient real fast for those who transact in bitcoin.
BitCoin is not backed by the trust of a government, correct. However it is supposed to be computationally hard to generate them. So its bales by your faith that someone hadn't made a big breakthrough in computer science.
(And I suspect even if you could magic wand it into existence it would still be very vulnerable. Take any fiat currency today of your choice. Remove the government backing. How long does it last? Probably not zero, but it probably crumbles at the next crisis at the latest. It would only take one metaphorical bank run to demolish the entire currency in a heartbeat. And it would only take a small nucleus of savvy people to prompt such a bank run, too, and such savvy people definitely exist.)
And in startup parlance, this is just about the worst possible "boil the ocean" plan possible. Virtual currencies will eventually happen and I'd even bet within a century you could live entirely off non-government-backed currencies, but it'll happen organically, not by fiat. And, no joke, it'll probably be some form of MMORPG cash, because those are by far the best seed I can think of for this sort of system to crystallize around, though probably some future MMORPG that makes today's look like tinkertoys.