[Aiwendil]

Here's a highly speculative idea: suppose we try to reconcile the various quotations by positing two distinct homes for Radagast?

At one time, Radagast lived at his 'old home', Rhosgobel, which was near the borders of Mirkwood at around Beorn's latitude (the northern of the two locations), thus satisfying:

It was Radagast the Brown, who at one time dwelt at Rhosgobel, near the borders of Mirkwood.

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."

Later (by the time of The Hobbit), Radagast left Rhosgobel for some reason and took up residence at another location further south, thus making this statement true when it was made:

perhaps you have heard of my good cousin Radagast who lives near the Southern borders of Mirkwood?

The only quotation against which this supposition seems to go awry is this:

...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. Radagast was not there; and they had returned over the high pass that was called the Dimrill Stair.

This suggests that Rhosgobel was at roughly the same latitude as the Gladden Fields. It also suggests that (even though Rhosgobel is his 'old home' and he is not there) Elrond expected Radagast to be at Rhosgobel. However, one could imagine that the scouts passed 'over the Gladden Fields' northward and so came to Rhosgobel (though why they would not cross at the more northern pass is a mystery). One could further suppose that Radagast had not definitively abandoned Rhosgobel in favour of the more southern location; perhaps he spent some time at each (and indeed, that may be why he was not at Rhosgobel when Elrond's people arrived).

I don't think I really believe the above proposal, but I thought it might be interesting to consider.