Slight correction: Vilya is the Ring of Air, and Nenya is of Water.
I think you have a very interesting idea there, Alfirin!. I have considered the connection of the Rings to elements as well, and here is what I think.
There are not
four elements found in most philosophies, but
five: Earth, Water, Air, Fire + something referring to the Spirit world: Aether or "non-matter" or "Void" etc.
The Greek Classical Elements (Earth, Water, Air, Fire, and Aether) date from pre-Socratic times and persisted throughout the Middle Ages and into the Renaissance, deeply influencing European thought and culture. The Roman Catholic Church supported the Aristotelian concept of aether because it supported the Christian view of earthly life as impermanent and heaven as eternal. The Hindu and Japanese also had essentially the same five elements: the four states-of-matter, plus a fifth element to describe that which was beyond the material world (non-matter or Void).
In Tolkien's Arda, Spirit World (Shadow World, the Invisible) exists parallel to the ordinary, physical, one (the World of Light) - like another side of one coin. So the concept of Aether would fit into the ME philosophy nicely.
I think that all the Rings may have been associated with each of the 5 elements:
Vilya with
Air
Nenya with
Water
Narya with
Fire
The Seven Rings all associated with
Earth
The Nine Rings all associated with
Aether.
And indeed, we don't even know whether there were noticeable differences either between the rings within the Seven or between those of the Nine. Therefore all the Seven and all the Nine could constitute one entity each.
Dwarven Rings surely made Dwarves think of Earthly matters: collecting gold and wealth, mining of mithril, maybe they also helped in the craft.
The properties of the Nine Rings, (barring preservation, which is the property of all the Rings alike) seem to be connected mostly with the Spirit World, with Aether. The Nine cause psychological effects on the wielders and the surrounding people.
So, no - no additional Rings were necessary. All the elements had their Rings.