I agree with all of this, but I want to add that it would be incomplete to say Gollum fell into the cracks of doom just because of the power of Frodo’s curse.
In Tolkien, words of magic have power because they reflect an underlying truth of the world. Gandalf says to the Balrog, “You cannot pass!” and indeed, the Balrog cannot pass; when it tries, the fragile bridge gives out under its feet and it falls.
Similarly, when Frodo curses Gollum, there is already an underlying truth of the world that explains why Gollum will be cast into the fire.
The one ring has consumed Gollum for so long that he no longer has any desire but to hold it again. And when at last he achieves his dream, Gollum is utterly fulfilled, no thoughts head empty, just vibing. This is a terminally dangerous state of mind when you’re standing atop a cliff overlooking a pool of magma.
If any other being in Middle-earth had seized the ring at that moment, in its place of power, they would have had something they wanted to use it for, and it would have been able to use them.
Only Gollum could have destroyed the one ring, because only Gollum was so wretched, so emptied out that he truly wanted nothing more than to simply have the precious. And so he didn’t watch where he was going, and he fell.
Frodo cursed Gollum, but, equally, the ring cursed itself. By robbing Gollum of everything that he was, it made its own destruction possible. In the biz, we call this a BIG OL’ WHOOPSIEDOODLE.
By the way: having Gollum, and by extension Sauron, basically slip on a banana peel and die is a genuinely hilarious punchline to 1,000 pages of angst. I’ll always be sad that Peter Jackson deleted one of literature’s sublime moments of comedy.