In LoTR Frodo prevails to Mt. Doom because of his desire to be rid of the ring, combined with the help from outside in the form of the new king of Gondor coming to challenge Sauron. (Of course even there he's unable to resist the lure of power and poor Smeagol has to take the dive). But Frodo has to go right into the heart of the enemy's power through danger and struggle and temptation.
That's a long long way from "do nothing, relax, and the problem will cure itself, man." That sort of approach was illustrated in LoTR (the movies, not the book) when Pip suggested they just give up and go home and leave everything for someone else to deal with. The closest JRRT himself gets to that is the bit in the Silmarillion where the 'Valar' call the Elves to Aman and abandon Middle-Earth in the first age (but really that sequence is more of a allegorical rejection of Deism IMO).
The 'nature will take care of what man cannot' is a theme in War of the Worlds, but don't extend it to Tolkein's work, please.
In the end, doesn't everyone fail the test, and evil is destroyed by grace?