The question of hobbits can be assessed through the analysis of individual hobbits. I will begin with Bilbo Baggins, and continue on to others.
Bilbo Baggins was a somewhat pretentious bourgeois, the son of the dominant Belladonna Took of the famous and powerful Took family, and he was interested in the things that reasonably well-off hobbits were interested in. The interests of hobbits at that time, and for the general span of hobbit history, were acutely local, rustic, and (for lack of a better term) gossipy. They were in the lap of luxury, not needing a real government to keep them in line (I believe that the Shire’s government would be a good example of the kind of “anarchy” that Tolkien wished for in his letters), and without any wide-ranging viewpoints. They simply did not concern themselves with bigger things.
After Bilbo was thrust into the situation mentioned in The Hobbit (see also “The Quest of Erebor”), Bilbo attempted to deal with the situations presented to him in strange and legalistic ways, ways which, in retrospect, look highfalutin and silly. But Bilbo proved himself to be very courageous, as he went in to see Smaug in his mountain, something the dwarves themselves did not have the stomach to do (it is worth noting that Bilbo maintained good relations with dwarves; they were with him when he left the Shire on his 111th birthday).
Bilbo came into possession of the Ring from Gollum, not exactly winning it fair and square, and kept it for the next roughly sixty years, until he left the Shire on his 111th birthday and gave the Ring and Bag End to his cousin/“nephew”, Frodo, son of Drogo, on his coming of age. Having the Ring for sixty years, Bilbo used it rather sparingly -- only on his adventures, or for games, or to avoid obnoxious people on the street. He never used it to try to manipulate the will of another person, but he kept the Ring with him at all times, and it grew on his mind as time went by. As Bilbo grew older and older, Bilbo began to feel stretched, “like butter over too much bread.” At this point, he was beginning the process of fading, by which a body which is mortal is clinging to the earth.
Tolkien believed that there was a difference between trying to cling to the world artificially and being one with the world, such as the elves. The Elves do not die a natural death, and their history is the history of the planet; but they envy the Men for their ability to die, and grow sullen and detached from the world the longer they exist. Yet for Elves this is natural: It was meant for Elves to stay on Arda. For others, this is not natural: the hobbit whose body clings to life will start a more painful and less graceful (note: graceful = full of grace) decline. Once Bilbo is separated from the Ring, he declines quickly into old age.
Frodo, however, is forced to bring the Ring to Mordor. He has more opportunity to be tempted, keeping the Ring on his person close to Mordor in a time when Sauron had all of his attention focused on it. Frodo and the Ring battle within himself, characterized by a battle between Sauron and Gandalf within Frodo’s own soul. As Frodo moved closer to Mordor, he met Gollum, and it is clear that Gollum is what Frodo can become, and shall become/will become if the Ring gets the best of him. In the Cracks of Doom, the battle within Frodo is lost, Frodo is both “broken” and “broken down” by the Power of the Ring, and he ceased to be the lively hobbit that Bilbo never ceased to be. He had begun the process of becoming Gollum, of becoming subservient to the will of the Ring -- and it was a blessing that the external Gollum defeated the internal Gollum of Frodo and stole the Ring, falling into the Cracks of Doom and unmaking the Ring. It is here that the process by which the substance of Frodo turning into transparent light ends, leaving Frodo a half unmade hobbit himself, part substance and part wraith, part hobbit and part otherworldly apparition. It is in this sense that Frodo is damaged; the process of becoming Gollum was not allowed to finish, and so, like an Alzheimer’s patient, he is both there and Frodo and gone and not Frodo. He has done his job. He has finished his quest for the Ring. And he will cease to do anything of much note ever again.
It is Frodo’s servant and body-man, Samwise Gamgee, who supports Frodo as he loses substance. It is by Sam’s will that Frodo physically goes to the physical brink, and I believe it is also here that he goes to the brink with the Ring, and falls. Frodo has fallen, but his body did not follow.
Later, Frodo, playing no role in removing Saruman from power in the Shire, removes himself from Middle-earth, going to the Gray Havens. He passes through, and will there “pass away” - even death exists for a hobbit in the Undying Lands. The poem in the Tolkien Reader begs the question, was Frodo able to enjoy his stay, or was he too corrupted by the Ring to be able to appreciate this sort of heaven-on-or-possibly-formerly-on-earth? The poem seems to me to hint that Frodo’s corruption has removed a part of him that is necessary to see anything but forest, the way that in a dream you can eternally search for something and yet never get any closer.
If Frodo died, then surely Bilbo did too. It seems that Bilbo, whose life has become one nap after another in his old age and decrepitude, will pass on, peacefully, without the torment in his soul that Frodo surely felt for the remainder of his life.
The Ring never really died until Frodo died.