Invisibility is incidental; it's a side effect for those who do not exists in the unseen realm. It was created to dominate those who had the lesser rings of power; the dwarves proved too resilient, and the elves took theirs off before they could be influenced. The ring is a tool of domination, and enhances the bearer's natural abilities. It may have allowed Aragorn for example, to inspire people to follow him as he became the next great dictator. It is a more subtle magic. In the books, Gandalf describes this process if he were to claim the ring; that he would build up a great kingdom to overthrow Sauron militarily and replace him as Middle Earth's tyrant.