Not really. Gold is held as a standard, but I would argue that gold itself is nothing more than another form of fiat. Why is gold worth anything? Because we all agree that it is. How much gold is there? Whatever is reported by the governments who hold it. There's no way to check. There are ways to guess, but it's just a shiny metal that we all agree is worth exchange.
Limiting us to gold also exposes us to massive economic swings, due to the limited ability to control how much or how little we all agree exists. I always hear people talk about how we 'print money' when we need it but NO ONE mentions that we also take money OUT of the economy all of the time.
"Why is gold worth anything? Because we all agree that it is."
True to some extent, but it does have more intrinsic value than paper currency. 1) It's rare enough that small amounts can be used to trade, but common enough that lots of people can have some. 2) It is pretty (subjective, but agreed-upon across many cultures). 3) It is chemically stable - doesn't rust or tarnish - so if you have 5oz today, you'll still have 5oz 10 years from now.
Reasons like these were explored on a Planet Money podcast, and they concluded that gold isn't an arbitrary choice; if you could pick any element from the periodic table to use for money, gold is the logical choice.
You described the requirements for a medium of exchange, and they make sense. But I don't think that that means "value" necessarily. Value is bread when I am hungry. Or gold when I need to make microchip contacts - but that usage is nowhere near justifying the price that gold actually trades at.
This is just a nitpick with the use of term "intrinsic value" - gold's characteristics make it a good form of money, but it's still (abstractly speaking) on a fiat basis.
I don't see how it could really work otherwise. There is no such thing as a good that everyone would value equally. Intrinsic value varies depending on the needs of the traders.
See, that's exactly the problem. The intrinsic value of anything - in this case, gold - relies on agreement of the scarcity. Truthfully, we don't know the scarcity of gold, or the amount that exists. We have only educated guesses.
Controlling the monetary supply with "paper" currency (which is an inaccurate term, "digital" currency is the reality) is much more stable and controllable. The US Government says they have $50,000,000,000,000, and the global economic capital markets either agree or don't agree.
Those are the only requirements for money.