[Mithalwen]

Radagast in his limited appearances in the books doesn't really seem either particularly qualified for or indeed up for much in the way of adventure. He is not a traveller save at great need and scarpers just about the very second his message is delivered. His journey to the borders of the Shire was surely via Rohan so he quite possibly had little or no knowledge of the mountain passes. Certainly unlikely to know more than the elves who live in proximity or those like Legolas and Gimli who presumably had come that way.

And he had inconveniently disappeared.

The western side of the Misty Mountains was more discreet and given the sense that is given that the ring has some control over its fate maybe it was thought unwise to take it back near to the places where it had effected two transitions of owner. Maybe too fanciful. But certainly the Western route tooke them relatively directly between probably the two safest places in Middle Earth protected as they were by Vilya and Nenya as well as the inherent strength of Elrond and Galadriel. The route had been scouted by the sons of Elrond and the first part quite likely had a measure of protection from the Rangers who still had settlements in the Angle.