I find it interesting that there's so much resistance to a natural mutation. The recipe for these color patterns were present in the DNA naturally. These sorts of things don't always just pop up overnight, either, you know; sometimes they are natural variants that birds carry in their genes for generations, a family legacy, if you will. These kinds of variations can actually be healthy and can be beneficial in the wild under certain conditions--sometimes a variant in color pattern or size will also affect something else, like disease resistance. For example, dwarfism has been shown to improve Marek's disease resistance in poultry. So, even a gene that might seem detrimental can be a positive variant under certain conditions, like drought or plague. Nature works in mysterious ways.
As for inbreeding to produce more specimens in captivity, what Monica says is correct. It is not automatically necessary to inbreed, especially if the variation is something that crops up in more than one specimen from time to time--and given the commonality of genes that dilute color or produce an alternate pattern in birds (this phenomenon is VERY common), I'd think it would not be too difficult to reproduce with only distantly related or unrelated individuals. And, by the way, just FYI, short-term inbreeding is not nearly as detrimental in birds as in mammals anyway. They naturally have a higher number of chromosomes (budgies, for instance, have, I think, some 31 pairs whereas humans only have 23) and more genetic variation. It is not the same as inbreeding a line of, say, dogs. By saying this, I am NOT advocating inbreeding; I'm just pointing out that a lot of our negative reaction to it comes from our experience with mammals, and it's not quite the same in birds.
