Basically, the Pseudohistory Channel or whoever you heard this from is simply wrong. Rather than a gold standard, the framers of the US Constitution tried to introduce a bimetallic standard - that is, a monetary standard based on both gold and silver. The Constitution states:
No State shall enter into any Treaty, Alliance, or Confederation; grant Letters of Marque and Reprisal; coin Money; emit Bills of Credit; make any Thing but gold and silver Coin a Tender in Payment of Debts; pass any Bill of Attainder, ex post facto Law, or Law impairing the Obligation of Contracts, or grant any Title of Nobility.
- Article I, Section 10, Clause 1 of the United States Constitution
While this applies to the States only, on the federal level, the Coinage Act similarly established a bimetallic dollar with both gold and silver coins. That efforts were made on both state and federal levels to introduce bimetallism demonstrates where early American preference lies.
The founders did not create a gold standard for the new republic, and thus there was no reason why they should have stipulated one in the Constitution.
This reflects the temperaments and philosophies of those who drafted the Constitution favouring bimetallism over either gold or silver singly. As the prominent economist Francis Amasa Walker wrote:
The third element of the silver party in the United States ... comprises the convinced bimetallists of the country; men who believe, with Alexander Hamilton and the founders of the republic, that it is best to base the circulation upon both the precious metals. These men are bimetallists because they believe that that system will at once avoid the evils of a strict money supply, secure an approximate par-of-exchange between gold countries and silver countries, and promote stability of value in the money of the commercial world.
- Walker, Francis Amasa. International Bimetalism. New York: Henry Holt, 1896.
The bimetallist champion, Alexander Hamilton, first Secretary of the Treasury, famously reported that:
Upon the whole it seems to be most advisable, as has been observed, not to attach the unit exclusively to either of the metals because this cannot be done effectually without destroying the office and character of one of them as Money, and reducing it to the situation of a mere merchandise.
- Report of the Scretary of the Treasury on the Establishment of a Mint. May 5, 1791.
A mono-metallic, gold standard was introduced only much later by the Coinage Act of 1873. For obvious reasons none of the Founding Fathers had anything to do with it.