The Wizards/ Istari were incarnated Maia (angelic spirits) who were sent to Middle Earth in about the thousandth year of the Third Age to resist Sauron. They were five in number: Gandalf the Grey, Saruman the White, Radagast the Brown and two Blue Wizards named Alatar and Pallando.
Most of what we know of the Wizards, as a group, can be found in the (mostly unpublished during JRR Tolkien's lifetime) texts collected in the chapter The Istari in Unfinished Tales. These texts include discussions relating to Radagast, and which are somewhat undecided about whether he failed in his mission - by turning aside from engagement with Elves and Men, and becoming 'enamoured' by plants and animals.
Recently, a further discussion of Radagast, written in 1970, was published in The Nature of Middle Earth edited by Carl F Hostetter, page 193:
[Gandalf] differed from Radagast and Saruman in that he never turned aside from his appointed mission... and was unsparing of himself. Radagast was fond of beasts and birds, and found them easier to deal with; he did not become proud and domineering, but neglectful and easygoing, and he had very little to do with Elves and Men although obviously resistance to Sauron had to be sought chiefly in their cooperation. But since he remained of good will (though he had not much courage), his work in fact helped Gandalf at crucial moments. Though it is clear that Gandalf (with greater insight and compassion) had in fact more knowledge of birds and beasts than Radagast, and was regarded by them with more respect and affection. (This contrast is already to be seen in The Hobbit 124-5. Beorn, a lover of animals, but also of gardens and flowers, though Radagast a good enough fellow, but evidently not very effective.)
This is somewhat damning of Radagast - as he is depicted as inferior to Gandalf in even his area of special expertise!
However, taking a wider perspective; I think we can make an interpretation of Radagast that gives him a great deal more credit for his activities in resisting Sauron.
In the first place, it seems likely that when Valar sent five emissaries, it is probable that each one (or pair in the case of the Blue Wizards who are considered as a dyad) - being angelic spirits each was of different nature and abilities; and these characteristic were no doubt chosen with the mission as priority. Even though Radagast was lesser in 'stature' among the Maia than Gandalf (as Gandalf was lesser than Saruman) - this does not preclude Radagast being better suited than Gandalf to Radagast's particular intended role.
Furthermore, each Wizard was probably affiliated with a different Vala; according to an idea developed by Tolkien in The Istari essay - which suggests that Gandalf was representing Manwe, Sauruman was of the people of Aule, and Radagast was chosen by Yavanna - whose special care was for plants and animals, and who in Middle Earth was represented by the Ents. I think we should infer from this that Radagast - despite his preference for dealing with plants and animals, and lacking confrontational courage - was especially well-suited by his nature and abilities for the task he was sent to perform.
In other words; although all wizards were intended to resist Sauron, each Wizard had a different specialist sub-mission. I suggest that Radagast's particular task was precisely to work with animals and plants to resist the attempts of Sauron to enlist them in his plans.
It is - after all - clear from The Lord of the Rings that Sauron had enlisted many birds and beasts in his service; especially as spies:
[Aragorn]: If the Riders fail to find us in the wilderness, they are likely to make for Weathertop themselves. It commands a wide view all round. Indeed, there are many birds and beasts in this country that could see us, as we stand here, from that hill-top. Not all the birds are to be trusted, and there are other spies more evil than they are... The Riders can use men and other creatures as spies... Soon I became aware that spies of many sorts, even beasts and birds, were gathered round the Shire... Regiments of black crows are flying over all the land between the Mountains and the Greyflood,' he said, `and they have passed over Hollin. They are not natives here; they are crebain out of Fangorn and Dunland. I do not know what they are about: possibly there is some trouble away south from which they are fleeing; but I think they are spying out the land.
[Elrond]: Soon now his [i.e. Sauron's] spies on foot and wing will be abroad in the northern lands.
Therefore it is clear that there was value in having a wizard who specialized in building alliances among birds and beasts - and presumably also among trees and other plants - in resistance to Sauron; and in opposition to the attempts of Sauron to enlist ever-more of the natural world in his evil schemes in hostility to Men and Elves.
And this - I think - was Radagast's special role.
If so, it is likely that Radagast was at-least somewhat successful; in so far as many (if not most) of the living beings in the North West of Middle earth (where Radagast operated) were Not on the side of Sauron; but cooperated with the powers for Good. The Eagles are one clear example; but most of the nature encountered by The Fellowship seems mostly-uncorrupted.
In sum: if Radagast is regarded as an emissary of Yavanna, with a mission primarily to the plants, birds and beasts (rather than Men and Elves) - then he may well have stayed true to this mission.
So much so, that (we are indirectly informed in The Istari essay) Radagast remained in Middle Earth after the defeat of Sauron - presumably because of his deep love for non-human 'nature'; choosing not to return over the sea to dwell in the Undying Lands.
Maybe, therefore, he is still here!