[Olog-hai]

That’s what they all say. But that would have put the Ring right in Sauron’s eyesight (the point was to keep it out of his sight); and the Sammath Naur, where the Ring was actually forged, was located within a side vent rather than the summit crater. Sauron did have air-to-air interceptors (the Nazgûl on their winged beasts) that would have kept the Eagles busy long enough to effect a recovery operation for the Ring, too.

Also, the book didn't specify whether there were any summit eruptions on Mount Doom; there could have instead been a lava dome up there instead of a liquid lava lake, and that wouldn't have been any use to destroy the Ring.

[Redcitizen]

They could have approached Mt Doom from the side opposite Sauron’s eye. Landed and snuck into the Sammath Naur. Tossed it into the lava and it’s adone deal. Then they would be home in time for mead/cornflakes.