These were deliberately struck for sale to collectors. Early 2001 articles in Coin World by Dave Bowers discuss practices by the mint in those years.
Unlike the 1851 and 1852 restrikes, these appear to have been struck much later. They are not known listed in either silver or copper until the 1870s. The Linderman collection had 2 examples of both restrikes 1851 and 1852 but not the 1853 which may mean these may have been struck in the time between his 2 tenures as Director of the U.S. Mint, ie when James Pollock was Director of the U.S. Mint from May 1869 to March 1873. In Ed Cogan's 11/1874 Sanford sale, the description notes: "Said to be only 12 struck".
Breen's listing for the silver proof in Woodward's 1864 McCoy sale has been discredited. It only realized $8.50 - much less than the prices realized for the 1851 and 1852 silver restrikes. The McCoy piece was purchased in Woodward's 1863 Colburn sale for $7. It is likely the same example which was offered in Woodward's 1862 Finotti sale.
Breen notes in his Encyclopedia of U.S. Proof Coins that these were struck using the regular proof reverse of 1862. Since the mint had at least 2 no-motto reverse dies in its possession in the 1870s, however, it is possible that pieces exist from more than just the mentioned reverse.
These were also struck in copper as J154/P183. Its first occurrence was in Haseltine's March 1876 sale.
Additional information can be found in "Silver Dollars and Trade Dollars of the United States" by Dave Bowers and Mark Borckardt.
Photo courtesy of Stacks.