I have to ask; do roosters get all randy with their mums/daughters/sisters or is there an inherent gene thing where they know to spread the gene pool elsewhere? Or is that not a problem?
It is a problem.
It's also quite complicated.
The chickens here tend to mate father to daughter, mother to son. Siblings are avoided where possible. It's a consequence of human keeping arrangements.
Here it tends to be mother and son, father and daughter. It's something that many breeders do not seem to understand.
Roughly, you can go 7 generations of generational inbreeding before one encounters problems. It's one of the reasons hatchery chickens are not a good bet.
The short term (maybe 50 years) is to only let the senior hen and rooster breed. The view where I live and among other people who are serious breeders is the seniors get to be seniors for a, or many reasons. Here the emphasis is on strong survivability and not on what looks nice.
So, if you have a senior breeding pair and you only let them hatch and raise chicks then that is only one breeding iteration. If the seniors live and breed for a decade then one can pick the survivors of their offspring and do it again for another decade. After 7 decades new genes need to be introduced.
There aren't many chicken keepers here on BYC with 7 decades of experience, let alone record keeping. The men that have kept fighting birds here probably have the greatest knowledge, far in excess of anything you can find on BYC. Because of the nature of cock fighting, breeders from all over the area get to meet other breeders and cocks and hens are swapped or bought. Some travel to other countries to get a particular chicken.
The stupidity of the American breeder and hatchery system is astounding and will eventually lead to irreparable damage to the chicken.
The above is a simplistic model but basically that is how the knowledgeable have done it for centuries.