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We simply do an experiment. You take your 100 paper dollars, and I take my 3 grams of physical gold, and then we head into a failed state with persecution, black market and a currency that is hyperinflating 200% every day and essentially is broken down. And then we see who can barter with what he has - you with your 100 paper tokens that after one hour already have lost significant value again and does so every minute and every hour coming, or me with my three grams of gold. I am extremely confident that my bucket will be several times as full than yours every time we do this.
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I would love to see this being done. Why? - would you ask. Because this implies that US is a failed state with currency that is hyperinflating 200% every day.
If this is not the case and people in the country in question understand what USD banknotes are (and we have to assume that for the experiment to be fair) then the person with cash is more likely to be better off, simply because it is more convenient to use money instead of the physical gold. I mean 3 grams is like a 1/6th of a cubic milimeter of gold, it is pretty hard to split it into 100 units to exchange it.
Regarding the rationality of that choice (apart from the convenience factor which is quite clearly there) - assuming that the USD is not hyperinflating and maintains it's buying power, then in that specific scenario there is little difference (apart from convenience, which is on the side of the USD banknotes), as what the person in question cares about is the value others assighn to the USD (and hence - it's purchasing power) which, the last time I checked, was quite good.
In fact in real hyperinflating situations (ie Russia in the 90s) foreighn currences were prefered to the gold and other such means.
Regarding gold - it's value is a subjective construct, if it was used just a regular good it would hold little value as there are few practical applications for it. The only difference between gold and cash (and other forms of money - I mean apart from metal and paper and now digintal money there were many, many others) is that it's value has been assighned by people for longer (which means little) and the money mass tends to not change rapidly (though we did see that happen with the other kinds of metal money - ie copper coins).
One could argue that this lack of change in the monetary mass (and the lack of convenience) is highly detrimental, as the money mass could not expand after the expansion of the economy, precluding the consumers from buying goods that they are otherwise capable of buying, as their demand is not supported by their liquid reserves.