Frodo shut and locked the round door, and gave the key to Sam. 'Run down with this to your home, Sam!' he said. 'Then cut along the Row and meet us as quick as you can at the gate in the lane beyond the meadows. We are not going through the village tonight. Too many ears pricking and eyes prying.'
(Frodo to Sam, Book I, Chapter 3, "Three is Company")
First of all, I doubt your premise is entirely true. I think you'd be hard put to find a place where Frodo said "I am going" when he knew he had companions. He rarely made a decision on which way to take, and never after leaving the Shire.
- First of all, Frodo had originally planned to leave the Shire alone, although he couldn't decide how or why, and kept putting it off.
- It was Gandalf who suggested Rivendell as a destination, and suggested Sam go with Frodo.
- It was Merry who led them through the High Hay into the Old Forest,
- It was Tom who rescued them (twice) and sent them on their way to Bree
- It was Strider who guided them from there, and finally
- it was Glorfindel who carried the unconscious Frodo into Rivendell.
I can put it no plainer than by saying that Bilbo was meant to find the Ring, and not by its maker. In which case you also were meant to have it. And that may be an encouraging thought.
(Gandalf to Frodo, Book I, Chapter 2, "The Shadow of the Past")
But even if your premise were true or even morally true, the mission is ultimately Frodo's. It is Frodo who has to get the Ring out of the Shire, and the other Hobbits (and eventually Strider) come along to help him get to Rivendell.