Mind if I ask what C.P. stands for?Some chickens don't synthesize their nutrient intake correctly so their eggs and meat will never match the best out there. Some of the very high production layers are not genetically predisposed to putting the full quality of their diet into their eggs even when on a more natural diet; however there's a few factors that affect this. I won't speak for backyard-reared or bred birds in this case, only hatchery-bought or production-keeper ex-layers.
Their eggs are much better once on a better diet but since there's a lifetime of 'catch-up' healthwise, they many take a year or more to lay truly healthy looking eggs. It's different once they've been raised minus the lack and damage of their ancestor's environments, and those hens I bred from high production layers were, by around the fourth generation, not noticeably different in quality from any other chook's. I see more American C.P.Layers looking red in the combs than I do Aussie ones and the genetic differences as well as feed makeup and vaccines etc is probably the deciding influence, I'd guess.
You could very well be right on the life time of catch up. I wouldn't know. I've only raised my layers (which are Red Sex Links) from babies. They were incredibly healthy. They didn't seem to lay any earlier than heritage breeds (much to my confusion), but the size of the egg they laid was very large. They were faithful layers. Friendly, and great foragers. I raised all of mine by broody.