an angelic being, with immense power.

The ring doesn’t turn someone invisible, but rather a physical being seen in the unseen world. Technically speaking, you could still see someone (like a shadow) wearing the one ring if stood still in full sunlight.

It could be theorized that Gandalf or Saruman likewise wouldn’t turn “invisible” because they are the same type of being as Sauron, part of both the seen and unseen world. Or maybe they could decide whether to be seen or not because of their nature and immense power.

A character not in the movies, Tom Bombadil, puts the ring on in the book and doesn’t disappear and he can see Frodo with the ring on, no problem. He also isn’t tempted by the ring either. Tom Bombadil is a character that almost nothing is known about, and there are many fan theories of who he is and his nature, but regardless, this shows that the ring affects and has different capabilities based on the ring bearer.

Lastly, yes I’m a LOTR nerd.