"You carry the fate of us all, little one."

[Imladris]

And most interesting to me (because I can't seem to reconcile it with my idea yet), why is it necessary for Frodo to lose the symbols of the other races and face the chasm naked and alone?

We must all fight evil, every single one of us. We can have help, but ultimately it comes down to us. Ultimately the individual decides whether he will give in willingly, or go down fighting. Nobody can help us with that at the bitter end. It is ultimately up to us.

Unfortunately, no one can win that battle, hence Frodo's defeat.