If the ringbearer died

[Evisse the Blue]

The whole point that Tolkien was trying to make was that Men, Elves, and Dwarves would have quickly succumbed to the power of the One Ring.

Actually, it is stated in LOTR that dwarves didn't succumb to the power of Sauron through the rings. The rings awakened the lust for riches in them, but that was all. However, I doubt one of them could have destroyed the Ring - although for different reasons than men, for instance.
It seems you're right - Hobbits were the chosen race.