You may find crossbreeding advantageous if you are seeking high egg laying birds for your flock. However, you only get predictable results in the F1 mating (meaning the first cross from a sire of one breed and a dame from another).
If you try to re-breed the F1 hybrids the DNA goes to chaos and you get 100 birds with 100 different qualities. As we say in farming, hybrids don't "breed true" meaning you get the same bird as the parents. This is only true of purebreeds. So you should always have a standing flock of purebreds available to you.
I personally think people ought to keep the same rooster as hens if they plan to breeds. Some breeds are endangered, so the more small breeders out there the better for the preservation of their genetics.
But with that said, purebreds have a certain look but aren't the most efficient birds as far as production qualities. A livestock breed is created by having a family tree that converges toa single point, rather than branches out. So, you're always working with a limited gene pool. So if you are farming for a hobby or profit, you'll probably want to be making your own hybrids.