Plato uses the story to make the case that man is tempted to do evil when consequences are absent and nothing is to stop them from doing so. The ring of Gyges functions in this manner as a device of temptation to do wrong. In short, it facilitates and encourages evil by its bearer.
In the same manner, Tolkien's ring constantly tempts its bearer toward evil. This functions well with Tolkien's overall philosophy in the book, which is strong in Augustinian platonism. Sauron essentially represents sin, evil, and a devil-like figure whose presence corrupts the world by alienating it from the good (evil for this purpose is defined as the absence of good or removal from the good). In contrast, various positive figures such as Aragorn, Gandalf, and the elves represent elements of the City of God - that which is good. The rest represent the worldly existence in between, with various degrees of alienation from the City of God and continual temptation from evil. They include the City of Man, which is tempted toward evil. Some within this realm have succumbed to the temptation of power toward that which is evil, such as Saruman and his armies. Others are tempted to the ring by their weaknesses, though they are disposed on the side of the good in the overriding conflict of the books. Many of the humans in Gondor exhibit this tendency. As for the ring, it tempts its bearer, no matter how strong, toward evil just as the ring of Gyges incited its bearer to comit acts of evil.
That's exactly what I always thought, yet I've never heard anyone else articulate it.
Thanks for the insight.