I've only had CCL's for a year and never an older hen, so they might go broody. I don't get a lot of broodies in my pens, except the bantam cochins and silkies. Even my BCM's have never gone broody. I did have a black Ameracuana and a Welsummer go broody earlier in the year, the Am set tightly, but the Wellie gave up after a week or 2.
Some breeds are bred for production, not broodiness, commercial leghorns are a great example. If you kept a large enough group of them, surely 1 would go broody, but I sure wouldn't count on it in a group of 20. Some breeds are known for broodiness (Orpingtons, Cochins, Marans, etc) but there are still hens in those breeds that never go broody, or give it a go for a few days and decide it's "not for them". There is certainly a genetic component, but also an individual personality involved. I think CCL's are not known for broodiness and it is generally regarded as undesirable, since they are meant to be layers, not mothers.
Even among hens that go broody often, some are much better mothers than others. We have some free ranging black hens that are fantastic. They hatch chicks somewhere in the hay and get them all down and out into the world with no help and rarely any losses. Then we have others that seem unable to account for all the chicks they hatch and only raise 1 or 2 in the end. I think our free ranging mixes tend to select for good mothering instincts, as well as aloofness around other animals (cows or predators). That is not at all what I select for in the pens as breeders, but the ones that I allow to escape to join the wild flock often revert to act like them quickly. Not always, some are lost of course, but it's rewarding to see happy birds running loose.