I guess I just don't see the Council of Elrond playing out like that. Frodo volunteered and the Council was opposed and had to be talked into accepting him as ring bearer. Gandalf evidently expected him to volunteer or thought he might, but the others didn't. I think if Frodo had not stood up and offered, nobody would have looked askance at him, because nobody expected him to or wanted him to.

At least that's how I read it, not to say it must be read that way.