OK, it’s on. Frodo did not fail, if we use the generally accepted model for success. Let’s assume that in next year’s Super Bowl the winning quarterback throws an interception in the final seconds as he is driving his team down the field for the winning touchdown. He is a failure, correct? But, suppose that the cornerback who intercepted the ball immediately fumbled it, and an opponent picked it up cleanly and raced for a touchdown just as the last seconds tick away? The quarterback will still be a champion, will he not? He did all he could, and only succumbed for a moment when he was tired and beaten up. I recall that Frodo was in a similar state when he fumbled. Hardly a true measure of his purity I would think. The failure of the body is not synonymous with the failure of will... I opine in your general direction.
Before he was sacked by Gollum, Frodo claimed the Ring for his own aloud. He was going to walk out of Mount Doom wearing the Ring of Power. As soon as he slipped on the Ring, Gollum hit him and then it was really on. Frodo lost the Ring to Gollum in the struggle and, exulting, Gollum danced over the ledge and into the Fires of Orodruin. Frodo warned Gollum that if he tried to take back the Ring, he would command him to be cast into the fire. As it happened, Gollum did fall in.