Let me preface this by saying that I'm very ignorant about this subject.
>What's the intrinsic value? Why is Bitcoin unlikely to be worth $0 in 10 years?
Why does a currency require a value as anything other than a currency to be expect to have value?
Bitcoin offers a useful service. There's is no reason to believe this service will stop being useful in 10 years. Therefore, it's reasonable to expect a demand for Bitcoin in 10 years, making it worth more than $0.
>Because it is spectacularly unlikely that gold will be worth $0 in 10 years
Isn't most of gold's value based on its demand as a currency? If people stopped buying gold just to trade and store value, wouldn't current gold owners lose immensely?