People keep citing forms of inbreeding being used as a method for "creating a line" of whatever as a justification for questions like that posed by the OP. What doesn't get mentioned is the other part of "creating a line" -- ruthless culling. See, when you breed relatives together, you're increasing homozygosity, which is another way of saying "bringing out the recessive traits" in offspring. There are good and bad traits that will pop up, and responsible breeders aiming to establish a line will expect to see poor offspring that would need to be culled, retaining only the best 10% or so for future breeding. Over several generations, you'll end up with breeder birds that start producing cookie-cutter offspring because the line has become fixed, but you will have to figure out how to "dispose of" all the rest before then. Is this something in your plans?
Yes, breeders will use inbreeding when trying to produce more of a rare new mutation, but the responsible ones will follow that with outcrosses to bring new blood back in. This is because there are two ways to reduce the chances of negative recessives from appearing in offspring. The first is what I mentioned above -- creating a line over several generations of breeding together relatives and ruthlessly culling anything that doesn't meet your standards. After a few generations, the negative recessives will have been mostly "flushed out" of the line, and the remaining breeder birds probably won't even carry those traits anymore. The second is to choose unrelated breeder birds who are less likely to share the same negative recessives, and thus their offspring won't show them. You might still get the odd poor-quality chick once in a while, but overall, you'll have a greater percentage of healthy and sturdy birds in every generation, even if they're more variable.
So, inbreeding in and of itself isn't bad. But if you're going to do it, realize that it's just one part of "breeding a line" -- the other being to expect inferior offspring at first, and the need to cull ruthlessly.