The One Ring and invisibility

[Galendor]

Hi all, I have a question that has bothered me for years: is invisibility a built-in attribute of the Ring's power, or more a random manifestation of its power that only occurs for certain wearers? If so, why invisibility?

I have only read The Hobbit and LOTR, in that order. Perhaps this question has been answered in other works of Tolkien. If so, please enlighten me. In The Hobbit, the Ring is presented more as a simple magic invisibility ring that Bilbo was lucky enough to find. Obviously that notion gets blasted into oblivion in LOTR.

As I understand it, the Ring's power depends upon who wears it - this is evidenced in statements made by Gandalf and Galadriel, wherein they refused the Ring because of the horrible power (and ultimate corruption) it would bring to them. I understand that the Ring's power is relative to whomever wears it, and that is why the Hobbits were able to bear it for relatively long periods. They inherently had very little "will to power" for the Ring to corrupt, so it had less immediate effect on them (correct)?

If I understand LOTR properly, it seems like the true power of the Ring is the ability to bend peoples minds to the will of the wearer, to subjugate them, and ultimately send the wearer back to Sauron.

So why does it turn Hobbits like Gollum, Bilbo, Frodo, and Samwise invisible? And their clothes, (but not their unsheathed swords, but that is perhaps for another post). It did not turn Tom Bombadil invisible, and these are the only people besides Sauron and Isildur who wore the Ring in LOTR.

If Aragorn had put on the Ring, would he have become invisible? What about Elrond, or Gandalf the Grey? Sorry I know there is more than one question here, but hopefully one or more of you Tolkien scholars can get me thinking correctly about the Ring and invisibility. Thanks!