The Rings are clearly magical, the clue's there in the title.
I wouldn't put it down to something as simplistic as them being 'power-ups' like a gem to be collected in a platform game; they are much more symbolic and complex than that. A Ring is already a powerful symbol - an endless circle which is worn on the body. The circle and the cross have been combined by the Celts and Saxons in their own representations of the Holy Cross. The Circle is cast by the pagan in ritual for protection. The circle is seen in megalithic remains, and it represents the cycles of life, the seasons, the planet.
We see the effects of the magical circle in Lorien, where Galadriel has created a beautiful and unearthly realm, protected from the wider world. She uses the magic of the Ring to create this place, almost like casting a 'glamour' over a part of the world. Elrond and Gandalf also use Ring Magic, as of course, does Sauron. When Galadriel 'talks' to Boromir and he is unnerved by her effects on his mind, this seems to be a clear example of Sanwe; she also seems to be able to reach Sam and Frodo in some way through her thoughts. There is also evidence that the control of thought was a factor with the Rings of Power when we look at how the Elves hid their Rings as soon as they 'became aware' that Sauron had his own power to discern them.
I see plenty of evidence in the texts that the Rings could have been used to enhance the abilities of sanwe which all sentient beings possessed whether they were aware of these powers or they were dormant. It explains so much.