What Is The Rarest US Coin Ever Minted?

Rarity and value are apples and oranges.

Issues with 1 known remaining example are clearly the rarest.

There are three of those.

How about the 1870 S Half Dime (unique), the 1870 S three dollar gold (unique but there may be a second in the S mint cornerstone.) Does an 1870 S quarter exist? they put a full set of coins in the cornerstone in 1870 and the quarter is the only one now that doesn't have a known example outside the stone. If they made duplicates of the Half Dime, Seated dollar and three dollar gold, they almost certainly did so with the quarter as well.

Not as rare as some of the others, but the 1853 O no arrows half dollar, no known mintage but 4 examples are known. One of which just turned up fairly recently. All are worn they did go into circulation.

I have no idea what that coin may be.
But one thing that I do know for sure:-
It will be a unique pattern experimental strike.

I would say the 1964 Peace dollar as well. I would also make a very strong case as to the legality of ownership. Based on mint records, 316,017 were struck beginning on May 12, 1965. In the first week of production Denver Mint Superintendent Fern Miller allowed employees to purchase or exchange for the new dollars, a usual occurrence at the mint. Twelve days later, May 24, the order from President Johnson went out rescinding the order for production. And here's the interesting part. It was at this time that Director Ava Adams sent a letter ahead of her arrival in Denver requesting that all 1964's obtained by mint employees be returned at once. Overseeing the melting operation, and again according to the official mint records, rather than do a physical dollar count, she decided to weigh them instead. Although no legal expert, they were authorized, struck, legal obtained and the mint's own financial books balanced after the fact. It's no different than the 1933 Ten dollar gold, which are legal to possess despite a Presidential Executive Order.

Or the 1894-S Barber dime .

If they did not count them, then they did not see every date on the surrendered coins either. Therefore, it is possible that at least one 1921-1935 Peace dollar was sacrificed in that melt. Possibly more. Many more. At least one 1964-D Peace dollar is out there!

Are we not going to restrict this to CIRCULATED business strikes?

There are tons of 1964 " Peace dollars". All Daniel Carr overstrikes. You can buy one on ebay right now: https://www.ebay.com/itm/184856876900