The enum approach for Singletons was popularized by Joshua Bloch in his book Effective Java. Another good way is the lazy holder pattern, which is sort of similar to OP's idea. I think hiding the enum within a class like OP proposes will not add any performance or concurrency risks.
Singletons are still used a lot, although they are often hidden within the frameworks we use. Whether or not to use a Singleton depends on the situation, I don't agree that they should never ever be used. Singleton has gotten a bad name because of the horrible overuse in some poorly designed systems.
