Why didn't Saruman go to the Shire to take the Ring?

[William Cloud Hicklin]

The only reason why Boromir and Faramir haven't figured it out as well, was that in Gondor they didn't know that Isildur took the Ring from Sauron. Sauron himself knew it fairly well. And he knew how Isildur fell. So the solution was easy.

Did he know how Isildur fell? Sauron was already neutralized by then, and would have had no particular knowledge of a skirmish two years later- and certainly wouldn't have any idea that the Ring, specifically, caused Isildur's death by slipping off at the wrong moment. The attacking Orcs didn't know about the Ring, and in the west only three survivors brought the news of the disaster to Elrond (only one of whom, "Ohtar," actually knew about the Ring).

For that matter, did Sauron even know that Isildur took the Ring? He was already 'dead.' PJ notwithstanding, it was Elendil and Gil-galad who did him in: Isildur was just looting the corpse.

In short, Isildur's possession of the Ring was known to very, very few: Elrond, the sole witness, and the survivors of the Gladden. Elrond's knowledge was circulated among the Wise, but is was always highly classified information.