Lets change your assumption and say they don't have to unrelated. That gets us back to reality. What you want is for them to not have genetic diversity in the traits you want but reasonable genetic diversity in the other things. This may sound like a strange radical thought but think about it for a minute. You create a breed or improve a trait by eliminating genetic diversity in a trait you want. Say you want a black bird. You want to eliminate any genetic diversity that would give you a red or white bird. That's how you make breeds.
But you want genetic diversity in the other traits. If you lose too much genetic diversity they can become sterile, non-productive, or unhealthy. It is kind of a fine line but people have come up with techniques to manage that.
One method is used by a lot of hatcheries, the pen breeding method. You are not going to get show chickens with this method, hatcheries are into mass producing chickens at a reasonable price. No matter how closely they follow the SOP when selecting their breeders this method does not produce a lot of show chickens but their prices reflect that. They manage to keep genetic diversity up by having maybe 20 roosters in a pen with 200 hens, just to pick some numbers. The random nature of breeding insures that genetic diversity stays at acceptable levels. They are all related but not that closely. The larger your flock the longer you can go with this method. Some hatcheries have kept their breeding stock healthy for decades with this method.
Breeders on the other hand are not into mass production. Instead of letting the chickens mate at random they select individuals that when mated are most likely to produce the chickens they want. They still have to hatch a lot of chicks and they have to feed and house them until they can select the best. Their prices reflect this. This might be show quality, it might be trying to improve a trait like laying or meat production, or maybe even create an stabilize a new breed. They may have one rooster with one or two hens specially selected. They use inbreeding to develop what they want (eliminate genetic diversity) in the traits they want to limit, but once they get there they use other techniques to maintain genetic diversity in other traits at acceptable levels. One common technique is spiral breeding. You can do a search to get better details but the basic idea is to set up three flocks, A, B, or C. The hens stay in the same flock for life. Every year you take the best rooster in one flock and cross it with the best one or two hens in another flock in a certain rotation, say A hens with B rooster, B hens with C rooster, and C hens with A rooster. Again they can keep their flocks going for decades.
One way this has been managed for thousands of years on small farms is that they keep their own replacement birds from the ones they hatch for a few generations. Again the larger the flock the longer they can go but even with one rooster and a dozen hens they can usually go four or five generations. Then they bring in a new rooster to reset genetic diversity and go through the cycle again.
Of course there are other methods and twists on any of these. I don't have a clue what might work best for you. Good luck!
