This is obviously a myth. If somebody mails a DVD of "Godzilla vs King Kong" to a government employee, it doesn't place the movie into the public domain.
Not a myth, in Sweden at the time it was the law, Scientology's "all your problems are caused because you are haunted by murdered space aliens" secret was put in the public domain this way
That is an interesting take. If somebody robbed a bank and gave the money to the state, the state would obviously need to return it to its rightful owner.
But copyright is not so straight forward. The church could contest that the physical copy belonged to them and was on loan. Thus it could not be gifted legally. The church could claim that the physical document was licensed non-transferably. They could claim that the copy itself was against copyright. There could be lots of arguments they could make to suppress the document. But none seem so straight forward as the bank robbery example.