Sauron created the Ring with himself as the only intended bearer. The Ring was imbued with the will and power to enslave its bearer, and I believe the invisibility it bestowed was a part of that -- pulling the bearer into the wraith world, like the Nazgul's rings had done to them. The invisibility of the Ring was not a good invisibility, in that it was not harmless: it was a shifting of the wearer into the wraith world. With the exceptions of Sauron and other beings with a strong enough will to possess the Ring rather than be possessed by It, I believe the One Ring would have had the same effect on its bearer as those rings given to Men. And once utterly enslaved by the One Ring, the bearer would have brought it to Sauron, and then died when he was separated from it. The Ring's bearer becoming invisible was, I believe, an indication that the bearer's will was weaker than that of the Ring.
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