On top of that, Elrond wouldn’t have been able to. There is a reason that Mt. Doom was undefended. Nobody could willingly destroy the ring in the fire, not even Elrond. The fact that it actually went in after the events of LOTR was pure luck (Edit: as someone else pointed out, it was direct intervention by Eru Iluvatar) that Gollum didn’t have very good footing.
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A well known, high ranking elf murdering the High King of Gondor and Arnor, just after the men came to the elves’ aid against Sauron. It would have sparked a massive war that likely could have meant the end of either of the races.