Beren wrote:
In the movie, Frodo and Gollum were struggling over the Ring, and they stepped too far, and fell. An accident.
The only difference in the movie is the fact that Frodo, after having his finger bitten off, decided to try to get the Ring back.
The point is right there in the second sentence of the quote. The destruction of the Ring in the book was the result of the direct intervention of God. In the movie it was because a pair of brawlers slipped and fell. One could suppose that
possibly Jackson intended for this to be the work of Eru (though personally I'm doubtful that he's ever read Letter 246), but it is still the intervention of
Frodo at the very end that makes it possible.
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And, visually, this is probably the best idea.
Looking at the scene purely from the perspective of visual interest then you are absolutely correct. However, I think that the story is more important when the two aspects have to clash.
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If Frodo was possessed by the Ring (which he was), that ownership that the Ring had on him would not end until the Ring died. So he wouldn't care about his finger for very long. The Ring's possession of him would draw him until the Ring itself was destroyed.
He was also probably in shock and intense pain, and it would take some time for him to be able to do anything. I think we're on the same page with this.
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In the book, this probably would have happened too, if it had not happened a lot faster. It was like, Gollum got the ring by biting Frodo's finger off, and he held the ring in his hand and fell off. In the movie, he rejoiced for a spell, so Frodo had time to act.
A lot of things in the book would've been different if the chain of events was altered. That's circular reasoning though.

Also, Gollum rejoiced in the book too, and it was during that rejoicing that he fell.
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Was it a good idea to add 30 seconds of time in the movie? I think it was. You get to SEE (which is what movies are all about) Gollum's reaction when he once again obtains ownership of the Ring. You also get to see the Ring's total possession of Frodo in the fact that not even a severed finger gushing with blood would stop him from retrieving his precious.
The movies could have SHOWN us Gollum's reaction and yet not deviated from the book. There were already plenty of examples of Frodo's obsession about the Ring and the weight it was on him, so that does not necessitate a change, particularly one with as radical an effect on the story.