Of course coin collectors and anyone dealing in bullion will not treat the coins as equivalent, and historically speaking this is one of the reasons for stamping the year and mint mark on the coins, but this is entirely at the discretion of the mint; there's nothing stopping them from issuing new "gold" coins with a lower percentage of gold and the same year mark, except for the same kind of social pressures that prevent the government from simply expropriating the contents of your bank account (as they did here in Argentina in 2001) or refusing to pay its bonds (as is happening here in Argentina right now.)
You might object that the United States isn't Argentina. But until just before 2001 Argentina wasn't Argentina either, not as we know it today; its troubles were behind it and it was the model country the IMF would show off to demonstrate how effective its economic recovery plans could be, even as the economy was gradually hollowed out. People living in the US, a country whose stock market just fell by 30% in a month and which didn't manage to produce or allow the importation of a reasonable number of COVID-19 tests until this weekend (although Korea, Japan, Germany, Taiwan, and China had no trouble with this) may find this sounds familiar.
If someone is concerned about the banking system expropriating their savings, it is not a reasonable response to offer them an alternative that makes it easy for banks, stockbrokers, or the government to expropriate their savings. That's precisely the kind of threat they want to defend against.
Heavy gold chains under your clothes are a lot safer when you're traveling than a bunch of coins. Moreover, the 1933 Executive Order confiscated the coins but not the chains.
> But why own physical gold? Why not just trade in gold futures or buy gold mining shares?
It's harder to confiscate, especially at scale.
> If you want the stability of the Swiss Franc, it can be traded on the foreign exchange market.
What does the franc have to do with anything? This is a non sequitur.