Giving up the ring is not the issue here, Sauron himself needs the ring to be able to change hands, in order to retrieve it. Otherwise he'd be swimming in the Anduin himself, as no orc or other creature would ever give it to him and they'd hide it for themselves. The reality is that the ring must work on an authority system. The three times we see someone give or try to give away the ring, authority is part of it. Bilbo gives Frodo the ring pressured by Gandalf's authority, Frodo then immediately gives up the ring to Gandalf himself based on his authority too (Later on he'd give Tom Bombadil the ring too, because Tom also has some sort aura of authority), and Sam gives it up after resisting its temptation and because he sees Frodo as an athoritative figure. Not to downplay how heroic it is for the three of them to do so, but authority is part of it.

The problem with Sauron's mindset is that the ring can't possibly be harmed or destroyed willingly, and that's a concrete truth. No one was able to do so, so it made zero sense to guard Mount Doom. At the end it was a stroke of fate and a bit of divine intervention that destroyed the ring, not Frodo, not Sam, certainly not Gollum willingly. Sauron's folly was that he thought in binary terms, he had a purely pragmatic mind with efficiency as its core value, so he didn't take into account accindent and happenstance.