* Having Frodo and Sam be simply by-standers at the conclusion of this epic journey, while Gollum beats them, takes the Ring, then luckily (for all the Good Guys) slips and falls into the lava thereby succeeding in their quest is a much worse ending than having them be active participants.
Their decision to re-write and re-shoot that scene is critical to acheiving a satisfactory conclusion to the quest. The version they decided upon also has the benrfit of further reinforcing the pull of the Ring (Frodo fighting Gollum for possession, dragging them both over, instead of him pushing Gollum off or Gollum falling off).