LogFAQs > #970729478

If I remember right, Cirdan of the Havens would have been there with Elrond and Isildur and I'm not even sure they were even there inside Mt. Doom. They most likely had any sort of conversation at some camp nearby. Regardless of where it was, surely Isildur, being a king, would have guards nearby. Even if Sauron was defeated there still could have been orcs or other creatures about.

Also, besides it not being in the nature of elves to just kill someone like that, Isildur was a distant relative of Elrond's, a decendent of Elrond's brother Elros. I doubt Elrond could kill someone akin to him.

'Alas! yes, said Elrond. 'Isildur took it, as should not have been. It should have been cast then into Orodruins fire nigh at hand where it was made. But few marked what Isildur did. He alone stood by his father in that last mortal contest; and by Gil-galad only Crdan stood, and I. But Isildur would not listen to our counsel.
This I will have as weregild for my father, and my brother, he said; and therefore whether we would or no, he took it to treasure it. But soon he was betrayed by it to his death; and so it is named in the North Isildurs Bane. Yet death maybe was better than what else might have befallen him.

Another thing to consider is if the wise of Middle Earth knew at the time that destroying the ring was the only way to keep Sauron from returning.

And while it's not thoroughly explained, Sauron losing the ring didn't happen like in the books. In the text it's implied that Sauron "died" while battling Elendil and Gil-Galad, who also perished, and Isildur cut the ring from Sauron's body, which was basically a shell at that point since Sauron's spirit retreated elsewhere.