Where did Radagast dwell?

[Legate of Amon Lanc]

Aiwendil, it's true, as you said yourself, that the version you posted does not make sense - however, what if we reversed it? What if Radagast at first lived near the Southern end of the forest, that was his "old home" - he lived there still by the time Gandalf visited Beorn (as Gandalf says to Beorn). This "old home" was what the scouts visited:

...and some of these had crossed the Mountains and entered Mirkwood, while others had climbed the pass at the source of the Gladden River, and had come down into Wilderland and over the Gladden Fields and so at length had reached the old home of Radagast at Rhosgobel

And Radagast was not there. He moved. Why? Because, as we read elsewhere, Dol Guldur was populated again, "and with power sevenfold". It was no longer safe for Radagast to dwell down there, so he moved further North and built "New Rhosgobel" in the land near the Beornings, who probably were his friends and where he felt safer.

And this:

In a very late note on the names of the Istari Radagast is said to be a name deriving from the Men of the Vales of Anduin, "not now clearly interpretable." Rhosgobel, called "the old home of Radagast" in The Fellowship of the Ring II 3, is said to have been "in the forest between the Carrock and the Old Forest Road."

is just how Christopher explains the note of his father. The note was, as it seems to me from this quote, something like "Rhosgobel was between the Carrock and Old Forest Road". Christopher added the words "called the old home of Radagast". Do you see where I am heading? Christopher was not aware that there were two Rhosgobels and identified the "old home" with Rhosgobel from this note. But the Rhosgobel from the note was Rhosgobel#2.

I'm not expecting this to be accepted as a canonical answer, but we have one proverb in Czech: I don't know if an equivalent is in English, but we say "So that the wolf was fed but the goat remained whole". I think this is the case when the wolf was fed - but the goat remained whole.