Just because gold and silver are referenced in the Constitution doesn't mean they're not arbitrary. If founders had added bronze or iron coins (like China had) or Platinum or Palladium it would have been just as arbitrary.
Right now the US Treasury and Fed jointly control the value of the dollar. With a gold standard, China and Russia would. That would be bad, actually!
Gold and Silver were recognised as money in the Constitution because the Founders recognised reality.
Money has a multi-part definition: requiring e.g. fungibility, the ability to act as a share of value, the ability to meet a coincidence of wants, and so on.
Only Gold, Silver (and plausibly Copper) fit all of the requirements of Money. Petrol doesn't. Ammo doesn't. Rhodium doesn't. Jade and cowrie shells and land don't.
Historically societies not trapped on desert islands nor ruled by tyrants have freely chosen to adopt Gold and Silver as money. The Founders recognised the vote of history.
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Right now the US Treasury and Fed jointly control the value of the dollar
Until the market says different. At 18% inflation, people are looking elsewhere. The collapse of the Euro will give the Dollar a reprieve, but when Saudi moves to the BRICS I can't see the dollar holding up for long.