All the Rings amplified the wearers’ base natures. The Elves were primarily concerned with stasis, that is, change seen as a bad thing. Both Ring holders put a protection around their realms so that nothing would fade and die. Gandalf has his nature as a persuader heightened.
The primary reason Hobbits were chosen to transport the Ring was because they really had no personal desires outside home and hearth. There’s a running gag in LOTR that people had heard of Hobbits but had no songs or stories about them. They never got around to make them known. This made them perfect for being less affected by the Ring.
When it is said that Sauron ‘held’ the rings, that doesn’t have to mean on his body. He could well have them in a treasure vault somewhere in his tower (and that seems like a good idea, actually).
In Tolkien’s universe power is a tricky subject. There is spiritual power and physical power. Just like in the real world, physical power is limited to the strength of flesh and bone. So, “powerful” warriors of human, dwarven and elven races are killed in battle or by trickery or by betrayal, just like any of us could be. Spiritual power is different. Galadriel was powerful in part because of her elven ring, but also because she had seen the light of the “Trees” in Valinor. The Trees were the precursors to the Sun and Moon that lit the world where elves lived among the Valar and Maiar (the equivalent of arch-angels and angels). That doesn’t mean that she would be of any use in a battle against men. In fact, I don’t think she would do well at all. But, in a fight against evil spiritual foes (Ringwraiths, Sauron, etc) she would be the most powerful in Middle Earth (until Gandalf returned after being slain by the Balrog in his true form as a Maiar).
Normal humans would not have the spiritual power, it is not in their nature. Aragorn was an exception, he was descended from the Numenorean kings. His ancestor was Elronds brother, also half elven, who elected to be counted as men. They also had an ancestor who was a Maiar, so he had that going for him as well.
I think the rationale is that their master’s power was slowly reaching its peak when the Black Riders were in the Shire. Later in the books, Sauron is sending out vast armies to contest the world, and the Nazgul’s power had grown apace.
Certainly not canon, but I used to a be a player of ICE’s long-defunct collectible card game, Middle Earth: The Wizards. In that game, the Nazgul could only attack with full-power in Darkholds and Dark Domains. If you send your party to, say, Minas Morgul, you opponent can hit you pretty hard with Nazgul. However, to use Nazgul to attack or menace a party in a more neutral territory, such as Bree or Weathertop, a player has to set up and play a sequence of cards to to use Nazgul to attack or menace a party. This leads to either a weaker attack, perhaps augmented with a Morgul-Blade to incapacitate a specific party member, or an indirect attack triggering fear or corruption.