Frodo could have refused the quest when Gandalf first, temporarily, laid it on him. He did not refuse, and he did not accept it because he thought he could better, temporarily, bear the Ring with more success than anyone else. He did it because he knew no-one who was obviously better suited when even Gandalf refused the task. And someone had to take on the job at once.
I see no signs of vanity in the description of Frodo’s thoughts or words at that time. Frodo’s wish is that Gandalf may find someone more suitable to take over the task in his place.
At the Council of Elrond, it is not Frodo who first offers himself, but Bilbo. The offer is first refused by Gandalf, and seemingly Elrond and the others in authority agree. Bilbo admits, “I don’t suppose I have the strength or luck left to deal with the Ring.”
Those present at the Council sit long in silent thought, Frodo among them.
… A great dread fell on him, as if he was waiting the pronouncement of some doom that he had long foreseen and vainly hoped might after all never be spoken. An overwhelming longing to rest and remain in peace by Bilbo’s side in Rivendell filled all his heart. At last with an effort he spoke, and wondered to hear his own words, as if some other will was using his small voice.
‘I will take the Ring,’ he said, ‘though I do not know the way.’
Elrond accepts Frodo’s offer to bear the Ring, and Elrond also says:
But it is a heavy burden. So heavy that none could lay it on another. I do not lay it on you. But if you take it freely, I will say that your choice is right.
In
Letters of J. R.
R. Tolkien, letter 246, Tolkien spends over 7 pages mostly related to this subject. Tollkien writes in part:
I do not think that Frodo’s was a moral failure. At the last moment the pressure of the Ring would reach its maximum – impossible, I should have said, for any one to resist, certainly after long possession, months of increasing torment, and when starved and exhausted. Frodo had done what he could and spent himself completely (as an instrument of Providence) and had produced a situation in which the object of his quest could be achieved. His humility (with which he began) and his sufferings were justly rewarded by the highest honour; and his exercise of patience and mercy towards Gollum gained him Mercy: his failure was redressed.
Tolkien continues for two more paragraphs, which you may read, ending this argument with:
I do not myself see that the breaking of his mind was any more a moral failure than the breaking of his body would have been – say, by being strangled by Gollum, or crushed by a falling rock.
I think that Ivriniel is indeed being too hard on Frodo, referring to all sorts of proofs of his thesis, but then mostly not providing those proofs.
Nerwen keeps asking for Ivriniel’s supposed evidence but Ivriniel just doesn’t present it. Galadriel55 gives a wonderful essay referring to Frodo’s actual thoughts from the text, despite not having her LotR with her. Pervinca Took indicates clearly that Frodo is generally not consumed with vanity.