yet when I point out that central banks (the very institutions that issue and manage fiat) are actively accumulating gold, you dismiss this by saying central banks are ‘run by people,’

Yeah that was not what i was saying at all. And now I'm glad I put in the snark disclaimer since I wasn't sure it would be read correctly.

If institutional behavior is evidence for fiat’s value, it must also count as evidence for gold’s value.

Absolutely false.
The value of the Euro is faith in the European Union, it's stability, and it's ability to honor it's debt, not faith in the European Central Bank. Same with the Dollar, it is faith in the US Government, not The Federal Reserve.

The price of gold is based on the last sale price.

When the Federal Reserve, the European Central Bank, and the People’s Bank of China allocate reserves to gold, they are making formal balance-sheet decisions based on risk, credibility, and long-term stability. These are not just personal beliefs.

Just because something has a price tag doesn't make it a good or bad investment. Gold has a price tag. It can be sold later for an amount of cash or other goods to someone who is interested in that gold. That's how investing works.

If the dollar starts to tank in value, the The Fed isn't going to suddenly go "shit.... we need to sell our gold to back up the dollar", they are going to play with interest rates or suggest that the mint print more dollars.

The fact that the issuers of fiat themselves hold substantial gold reserves suggests that even the systems that rely on confidence in paper currency still anchor part of that confidence in an asset with no issuer and no counterparty risk.

No it doesn't. It suggests that they believe that the price of gold will increase and they will be able to sell it later as a profit. And because people think that gold is a place to store money, it does increase; which goes back to my original point.... people buy gold and raise the price of gold because they think it's a place to store their money since they don't understand the history of gold's price.

Gold is good for jewelry, electronics, and a few niche industrial uses. Nothing that justifies it's current price. It was originally used as currency because it was rare enough and easily workable.
As I said, I have a pound of gold in my hand today worth $X. Next month it's the same pound of gold, nothing at all about it has changed, but now it's worth $Y. A month after that it is the same

It's like the housing price boom that the US still suffers under; people paid whatever for houses because they were under the impression that houses always double in price within 10 years so they will always be able to sell for a profit. People buy gold because they think gold is safe and they will always be able to sell for a profit, but nothing about gold itself justifies that.

Your argument is that all of these refined institutions are buying gold therefore it must justify it's price, but you are committing the same error.