[Nils]

If Gollum had not died, Sauron would have recovered his ring and the quest would have been lost. There is no way anyone could willingly cast the ring into Mt. Doom. The only thing that would have happened is that either Frodo or Gollum would have faced off with Sauron and been defeated.

If Sauron were to recover the ring in such a contest, I believe that would have been worse than falling into the lava.