That it allows Sauron to find Elrond, Galadriel, and Gandalf?
Are they hard to find? I know Galadriel and Elrond's addresses. That whole trilogy can't all be just because Sauron couldn't find Gandalf, surely? Ironic, if so, since he was accompanying the one ring that could find him.
No - but then Gandalf wasn't even around in the East when the One Ring was forged, if I recall correctly.
Sauron wanted to dominate everyone in Middle-Earth, really. He wanted to do that through the leaders of people. So, he put on a pretty elven face, called himself "Annatar", the "Lord of Gifts", and brought them this whole story about how rings would be awesome. He taught Celebrimbor (an elf, who we don't see in the trilogy at all, as he's dead by then), a whole bunch of stuff about magic rings.
Celebrimbor made 16 rings (the Nine for Men, the Seven for Dwarves) with Sauron's assistance. Celebrimbor made the Three without Sauron. Thus, the leaders of Elves, Men, and Dwarves *all* had rings. Sauron then went and made the One, to try to dominate *everyone*, through those leaders.
Since the Three weren't made by Sauron, the elves at least had a chance to avoid that fate - they could sense the attempted influence and take their rings off. At this time the Elves with Rings were Gil-Galad, Cirdan, and Galadriel. Gil-Galad handed his off to Elrond, which is good. There's a bunch of warring that goes on (Sauron captures Celebrimbor and tortures him to death trying to find where the Three are, Gil-galad is eventually killed by Sauron on the slopes of Mount Doom, and so on). Cirdan, Galadriel, and Elrond just basically sit on their rings until Sauron is apparently defeated by Isildur, Elrond,
et al.
Eventually (if I have my dates right, about a thousand years after Sauron got the One Ring cut off his hand), Gandalf shows up, Cirdan recognizes him for what he is, and gives him his ring. It is another milennium before Smeagol/Gollum finds the One, and another few hundred years before Bilbo meets Gollum - so Sauron's without his ring for something like 2500 years.