For your first point - the supply of gold is hard to increase, whereas fiat money can have its supply increased infinitely at the whim of a government, or significantly each year by the commercial banking system. That's a major difference, given the impact that supply has on price. As for the second point - this one is obvious. Apart from fiat currency, gold has by far the best collection of properties beneficial for a monetary substitute - it's high value per oz so relatively easy to store and transport high amounts of money, it is very durable (high melting point, low reactivity), it is relatively easy to measure its purity, it is fungible/homogenous (unlike sea shells), it is attractive and has reserve demand for jewelry, and it is time-tested as a monetary substitute. If fiat currency falls from favour, gold is the only serious monetary substitute out there at the moment.
Well, this is very touchy-feely, if you ask me, so the rational person in me simply refuses to accept this as a justification. However, the irrational pre-historic hoarder in me definitely agrees. So I own some... Yep, true dat...
True that, but I am actually concentrating on the demand side. How can we blithely assume this arbitrary demand for gold continues ad infinitum? If you assume that demand for fiat money can come and go, why should we necessarily assume that the demand for gold behaves differently? I am familiar with all the "nice" properties of gold. However, there are all sorts of other materials that possess similarly advantageous properties or are even better. So, ultimately, are we just saying that "we love gold, now and always, because we have always loved gold (because it's shiny)"?