system
Given the following post, respond as if you're commenting on Reddit:
human
[Lord of the Rings] Why does the One Ring make Frodo and Bilbo invisible, but not Sauron?

For that matter, why does it provide invisibility at all? It seems a rather odd and irrelevant power for the ring designed to control all the others. “This here’s the Master Server. From here, with the right admin credentials, you can control any computer in the building’s network. Every piece of data from the outside flows through here. Oh, and it also makes toast.” I mean, that’s great, but... why?

It’s been far too long since I read the books, so I’m basing this off the movie (which I’ve paused to write this). In the opening montage, we see Sauron march into battle with his enormous mace, completely visible, but he’s wearing the Ring.
gpt
Sauron is a Maia, one of a race of powerful pure unseen beings - his body is in many ways a construct to allow him to interact with the physical world. The One Ring contains part of his power, and as a side effect shifts you to his world.

So the invisibility is an untintended consequence of what the Ring is, but it wouldn't have that effect upon Sauron since there would be no "shifting" of him to the Unseen "wraith world" because he's already from there.