Can a trait be too successful? Wouldn't overwhelmingly successful traits limit variability, which is one of the requirements of NS?
Yes, in neutral theory of evolution this is called fixation - when all the alleles (for a given locus) become identical. Note that it may happen, even if the gene does not have a striong selective advantage - due to the random effects.
If you are looking for the basic background in population genetics, Gellespie's book is a good starting point.