Hi Rachel and Odola,
I am not sure that the swift dismissal of Bilbo's volunteering is an important question? I do think it is a curiosity, however.
One of a number of curiosities in the very curious chapter, 'The Council of Elrond'.
Boromir's Divine Dream (which should have been the most important 'message' to the Council) seems to have been swiftly dealt with and forgotten. Note Gandalf's comment, (in the beginning of the next chapter) after Bilbo said, "Everyone had an eye-opener. Even old Gandalf." and Gandalf replied, "If you want to know, the only real eye-openers, as you put it, were you and Frodo; and I was the only one who was not surprised." Really Gandalf? Boromir's Divine Dream did not surprise you? How come? Or have you dismissed and forgotten it?
The exact outcomes expected from throwing the Ring in the Fire are never stated. It is certainly never clear that throwing the Ring in the Fire will destroy Sauron and win the War! It can be deduced (though again never stated) that throwing the Ring in the Fire will destroy the Nazgul and weaken Sauron (and answer the Divine Dream), but it needs to be deduced and the clues are well concealed.
The impression of the Council, is that Gandalf and Elrond are concerned only with the disposal of the Ring (more-so than with the War, or with Boromir's Divine Dream). They also seem to have pre-conceived their desired outcome, and to be steering the Council towards it.
Now it may be (as Prof Olsen has argued) that they detect the currents of history, the devices of Providence, and the progression of the Song, and consider that their desired outcome is what fits. However, this is quite a stretch interpretation to expect the first-time reader to come up with?
I think the first-time reader, if reading critically, rather than just swept along by the narrative in willing suspension of disbelief, must be somewhat baffled by the entire Council. What exactly will destroying the Ring accomplish? Does this counsel answer Boromir's dream? How? What makes Frodo such a preferred candidate as Ring-bearer? etc. etc.
I could see a careful first time reader wondering, "Are we setting up a tragedy here? Are the prejudices and fears of Elrond and Gandalf going to lead the whole enterprise astray? How will Frodo ever be able to fulfil the mission? He could not throw the Ring into his hearth fire, after seeing that it did it no harm. Why would anyone think he can throw it into Mt. Doom? How possibly could the Ring then come to be destroyed? What will happen after the Ring is destroyed? How will Sauron then be defeated? Will he be defeated?"
"Or, is this all going to end in some Gotterdammerung of all the Free Peoples?"
"Well, I guess, read on, and we will find out!"