I guess my goals for keeping birds are different from others, and that's why I'm not a fan of inbreeding. If you are seeking to produce birds with certain characteristics with regularity, then inbreeding would be used to create a line of birds that produce birds just like them. Just be aware that lots of things are affected by inbreeding, including fertility, immune system, longevity, susceptibility to cancer, etc. and not just "type." In the beginning of creating a line, you'll have to do lots of culling to remove homozygotes for harmful or undesirable recessive traits. In the end, you'll start to produce cookie-cutter birds. For people interested in showing birds, this is a goal. For people interested in maintaining captive populations of wild species, this is to be avoided. My mindset is from more of a zoological perspective. The species I'm interested in keeping would be bred with only unrelated individuals so as to maintain vigor and long-term population health. I prefer variability. If you prefer to consistently produce birds of a desired type, then inbreeding is your tool. If you prefer to keep a species in captivity and maintain genetic diversity, then inbreeding is to be avoided.
Also keep in mind that when beginning a line of birds, you must begin with truly exceptional individuals. Simply taking any old ordinary bird and inbreeding it will give you lots more ordinary birds. The goal in creating a line is to perpetuate exceptional traits that reproduce consistently. The OP isn't stating that this was the original goal. Rather, what I'm gathering is that the inbreeding is happening simply because he hasn't gotten any other stock.
Also keep in mind that when beginning a line of birds, you must begin with truly exceptional individuals. Simply taking any old ordinary bird and inbreeding it will give you lots more ordinary birds. The goal in creating a line is to perpetuate exceptional traits that reproduce consistently. The OP isn't stating that this was the original goal. Rather, what I'm gathering is that the inbreeding is happening simply because he hasn't gotten any other stock.