Leaving aside the idea of divine providence (as I've nrver been good at such theorizing) I'd suggest that Gandalf was to a great extent 'responsible' for Frodo's decision. Remember, Frodo had always taken Gandalf's words at their face value. To Bilbo he says:
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You cannot take this thing back. It has passed on.
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To whom? Everyone present had been told that. So what was Frodo supposed to feel, as no-one stood up? Could he help feeling it was HIS quest?
And besides, Elrond's sugnificant remark about SMALL HANDS doing great deeds. Thus Frodo, naive and sensitive, was led to making his decision. By no means do I underestimate Frodo's desire and eagerness, But to me things look pretty much decided beforehand. And, it turns out, for common benefit.