'not a moral failure'
Frodo's statement immediately before he claims the Ring has always seemed significant to me. It strikes me as being very carefully and precisely worded, and when I'm reading Tolkien I take that as a signal to pay attention to exactly what's being said. And what Frodo's saying, exactly, is 'I do not choose now to do...' which is very different from saying, 'I choose now not to do...' It's the difference between "I do not choose" and "I choose" - which is a pretty strong difference. What it says to me is that Frodo isn't choosing anything at that moment; the Ring is in control. Frodo literally does not choose. (This is the one place I dare to disagree with Christopher Tolkien, who said he doesn't think the revision from 'I cannot do' to 'I do not choose now to do' is 'very significant.')
When JRRT's Letters were published, it seemed to me they supported this interpretation. While Tolkien might not have parsed the specific line, he's very clear that this is, indeed, Frodo's situation at the Sammath Naur. The power of the Ring is too great for him. Letter #246 compares Frodo at this point to someone who has been crushed by a boulder. Because he's incapable of making a free-will decision, Tolkien says that Frodo's failure to destroy the Ring 'was not a moral failure.' What Frodo does, or doesn't do, is not a choice. Literally, 'I do not choose now...'
In another Letter (#181) JRRT compares Frodo at this point to someone whose will has been broken by brainwashing. If the tormenter does a "good" job, that person won't be aware that his own will is gone but will believe that his will is what the person doing the brainwashing tells him it is. That's the whole point of the process. What Frodo experienced at the Sammath Naur wasn't "temptation" (from the way I read it, the Ring hadn't been able to actually tempt him since Cirith Ungol, since I believe temptation needs to carry an illusion of something good - such as Sam's image of a garden swollen to the size of a realm as he wears the Ring); it was complete domination of his will by a stronger one - Sauron's, which the Ring carries. The battle of wills had been waged constantly throughout the entire trek across Mordor, and at the Sammath Naur the stronger will finally wins.*
Even then, Tolkien speculated in Letter #246, if Gollum hadn't interfered it's possible that Frodo could have struggled back until his own will was enough in control to be able to cast himself - with the Ring - into the fire. (Tolkien writing as historian couldn't say this would have happened, of course, only that it might have.) This brings up a question where I have no evidence of author's intent: If Frodo believes he wills what the Ring does, why doesn't he say, 'I choose now not to do...'? He thinks he's choosing, doesn't he? Is the phrase 'I do not now choose to do...' simply a clue for readers, or is it intrinsic to the story? In order for there to be even a slight possibility that Frodo might have regained control, would it have been necessary for him to have had, somewhere deep inside, a fragment of identity that was still Frodo? Not enough to keep the Ring from forcing its will on him, but enough that somewhere (possibly only subconsciously) he sensed that 'I choose..." would be an untruth?
There are some things Christians tend to say to each other that Tolkien forces us to look at more deeply - or from a different direction - than we might otherwise. One of these is that God will not allow us to be tempted beyond our ability to resist sin. If we honestly look at the way life works in our world, we may need to go beyond the obvious meaning of that truism if we want it to be true. As Tolkien said, comparing Frodo's situation at the Sammath Naur to that of a person broken by torture or brainwashing, there are times when a person is placed in a situation that's simply beyond that person's ability to cope.
What does this do to our truism? Can we still trust it? And the question behind that question: Can we still trust God? My answer would be "yes," as long as we recognize the difference between committing a sin and doing something wrong. In claiming the Ring, Frodo was definitely doing something wrong, and God had allowed him to be placed in a situation where it was beyond his ability to resist doing it. Tolkien heard from one reader who said that, instead of being honored, Frodo should have been hung as a traitor; if you look at only his action, you could make a case for that. But sinning requires a freely-made choice. Although Frodo not only failed to destroy the Ring but even claimed it as his own, Tolkien says this was 'not a moral failure,' as Frodo's free will was completely overpowered at the crucial moment. Looked at this way, our truism becomes a foregone conclusion: If a person is in a situation where he or she cannot resist doing evil, there is no sin committed. Although God may allow someone to be forced to do something evil, God's mercy and justice protects that person from sinning.
I've avoided using "we" when talking about that kind of situation because I don't think it's common. Most of us need God's mercy a long time before our straits are that dire, and if we're honest with ourselves "The Devil made me do it" is seldom a complete defense. When Tolkien discusses it, he limits it to extreme circumstances (see letters #181 and #191). He does, however, see it as a reminder of our general human limitations: 'But one must face the fact: the power of Evil in the world is not finally resistible by incarnate creatures, however 'good'; and the Writer of the Story is not one of us.' (letter #191) Frodo resists the power of Evil to the extent of allowing his identity and personhood to be stripped from him. His loss of free will at the end shows that he's given everything, including his very self, to that resistance. But even that wasn't enough. Our inability to finally resist the power of Evil in the world is the reason we need a Savior, not just personally but as humanity, something the Writer of the Story knows even better than we do.
*In an earlier draft of the book, Tolkien put the reader within Frodo's mind at the Sammath Naur. In that draft he is being tempted, by being shown all the good he could accomplish if he used the Ring. But what we see through Sam's eyes in the story as published seems more primal than that. As with most of the revisions he made along the way, I believe Tolkien made this one for a good reason. And, very importantly, in his letters following the book's publication, Tolkien never refers to the incident as a temptation - any more than someone could be tempted to have a boulder fall on him. In his most extended discussion of temptation, in letter #181, Tolkien says of someone who is caught in a situation that demands "a strength of body and mind which he does not possess," that he is "doomed to fall to temptation or be broken by pressure against his 'will': that is against any choice he could make or would make unfettered, not under the duress."