So my question is not why does it make you invisible, but actually, Did Sauron want the One Ring to do that? If so, why? What purpose does that serve?
Good question.
Yes, the Ring brings you into the Unseen World, the Wraith World, and thus turns you invisible for everyone who only dwells in the Seen World.
But why he does that is something, I don't think I ever heard of. ^^
I think Galadriel mentions, that someone who could use the ring, wouldn't turn invisible. So like VectorMaximus mentioned, only People without a strong Will or magical powers will turn invisible "by default".
(though I would actually say Isildurs Will might have been strong enough. Same goes for Aragorn btw.)
So my
speculation would be that the invisibility part was just another unplanned side effect of the ring, that Sauron never expected that or used it in that way.
As you said Hamanathnath, I also think it doesn't make sense for Sauron, who could have controlled and beaten enemies with the ring, to use it for hidden attacks or stealth missions.
After all, we know of Saurons greatest manipulative Mission: The Seduction of Numenor. And he did it definetly WITHOUT the One Ring.
So why does the One Ring Transport you into the Unseen World? What is the Unseen World anyway?
No idea.
Maybe it is a world where Spirits are much easier to access? Since the One Ring gives you the power to dominate minds, maybe he's channeling your power into the Unseen in order to dominate other Spirits from there?
Maybe. Who knows... ^^
So if invisibility was built into the ring back when he could be fair, it could also serve as a way of him escaping defeat.
Back when he was fair, means also back when he could take any form he wanted. Why should he Need to turn invisible then in order to flee, when he just as well Change his form into something unnoticeable, or maybe completely leave his Body behind?
(though I'm not 100 % sure if Sauron could have leaft his Body...)
Sauron loses the Ring because he gets cocky off his victory over the 2 High-Kings, personally. He's not expecting Isildur or anyone being able to defeat him at this point. That's why he goes to 'humiliate' his enemies in killing Isildur by choking, to serve as an example of his power.
Nope.
Just in the movie version.
In the book Sauron lost in an epic brawl against Gil-Galad AND Elendil AND Elrond AND Cirdan AND Isildur, all at once. Sauron was already lying defeated on the ground (after killing Gil-Galad and Elendil) when Isildur stepped up and cut the One Ring off his finger.