What is clear, is NO ONE at the Council thought any of the
hobbits were appropriate 'material' to carry
the ring to begin with.
Not a one was even mentioned as a candidate.
Others were... as well as other solutions like pitch it into the sea. Therefore, the mindset of the council - every member of it - was not in the least turned toward any hobbits.
Dwarves,
Elves and
Men were rejected. Even the
Valar were.
At the end of the day,
Bilbo offered himself on the chopping block... (not the council)... and the council (especially
the wise)... easly saw he was not the appropriate one.
But when
Frodo offered himself... the wise saw Providence brought it to this point, and Faith would see it through.
To the thread's title, why was Frodo chosen? Because no other candidate had been chosen by
Eru to do the job. And in spite of sight, faith saw the way.
It need not be any more complex than this.
Had Frodo proven himself to be a mighty anything up to that point, making him a candidate? Not in the least. He had, indeed, failed to prevent them from getting too close to the Downs, failed to prevent himself from being muddled, failed on the roadway, etc. etc.
But his qualification for the task had nothing to do with himself...
It's true, in many ways hobbits aren't given their due. But to the Council, hobbits are nearly the 'unknown stuff of legend.' Even the wise (save possibly
Gandalf) have little real understanding of how 'tough' a
hobbit might be to resist the ring, etc.
We need to be sure to divorce ALL we know about hobbits by the end of having read the story, from what little was known about hobbits up to the time of the council... and what was known about them 'to the members of the council'... not to ourselves.
Then we can more rightly see 'why Frodo was chosen.' They were clueless 'how tough a hobbit might be to be able to resist the ring' (almost) etc.
Like the issue with 'Christianity' vs. 'elements of religion' we need to put ourselves in the shoes of the characters in the text, not in the shoes of the readers.