Why Man?

[The Squatter of Amon Rûdh]

The Silmarillion has something to say on this:

Seven rings he gave to the Dwarves; but to Men he gave nine, for Men proved in this matter as in others the readiest to his will...The Dwarves indeed proved tough and hard to tame; they ill endure the domination of others, and the thoughts of their hearts are hard to fathom, nor can they be turned to shadows.

It goes on to say:

Men proved easier to ensnare. Those who used the Nine Rings became mighty in their day, kings, sorcerers, and warriors of old. They obtained glory and great wealth, yet it turned to their undoing. They had, as it seemed, unending life, yet life became unendurable to them. They could walk, if they would, unseen by all eyes in this world beneath the sun, and they could see things in worlds invisible to mortal men; but too often they beheld only the phantoms and delusions of Sauron. And one by one, sooner or later, according to their native strength and to the good or evil of their wills in the beginning, they fell under the thraldom of the ring that they bore and under the domination of the One, which was Sauron's. And they became for ever invisible save to him that wore the Ruling Ring, and they entered into the realm of shadows.

(Of the Rings of Power and the Third Age, p.289 of my edition)

My reading of this is that it wasn't long life that turned the bearers of the Nine into wraiths, but the domination of the rings themselves, and the influence of the One acting through them. Of course it's still possible that this is a case of the sword outwearing its sheath, but there's no indication of a time-scale. They were also using their Rings regularly, and for greater things than achieving invisibility for short periods, probably hastening their collapse into thraldom.

As for Hobbits, we have an implication from Gandalf that they may have been made of sterner stuff than Men:

Soft as butter they can be, and yet sometimes as tough as old tree roots. I think it likely that some would resist the Rings far longer than most of the Wise would believe.

(The Shadow of the Past)

Perhaps a combination of this trait and the fact that Gollum didn't use the One for great things, or particularly often was what kept him from becoming a wraith. I think that the same thing can be said of Bilbo, and the other Hobbit Ringbearers carried the One for only a short time, and used it hardly at all.

Of course, there's always a possibility that the One was never designed to turn its bearer into a Wraith, or that it was a property of the Ring that required a specific act of will from Sauron, which would still be consistent with the piece quoted above. I'd be interested if anyone could shed any more light on this one.