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No, you do not need a pure breed. As far as chickens go, pure breed does not mean a lot anyway, depending on who you get your chickens from. Certain breeds should have certain tendencies, but not every breeder selects their birds for those tendencies. For example, Orpington should go broody a lot. Enough breeders do select for them to go broody that Orpington are often a good broody breed. But when a hen is broody, she is not laying eggs. Certain commercial operations eliminate from their laying/breeding flock any hen that goes broody. They are taking up space, eating expensive feed, and not laying any eggs to pay their way. After a few generations of only hatching eggs from hens that don't go broody, you have a strain of birds that does not go broody often.
Another example that may help explain it. I recently saw an article where a breeder took a flock and divided it in half. He selected breeders from one of the flocks for big size and the other flock for small size. The article did not say how many generations it took, but the average size of birds in one flock became 9 times bigger than the average in the other flock. These were purebred birds from the same ancestors. To me, that shows the power of selective breeding.
There are a lot of misconceptions about crosses versus pure breeds. If you have a flock that has been selected to lay well, you will get offspring from those parents that will also lay well. It really does not matter if those are purebreds or crosses. And it does not matter if they are sex links or not. Several hatcheries sell those commercial laying sex link birds so the general conception is that sex links are egg laying machines. If they are from the commercial breeds, they are. But if they are made by crossing regular breeds, not the highly selected commercial types, they will not lay any better than the parent breeds. These parent breeds at the hatcheries lay real well too, so practically everyone is happy with the sex links they get from the hatcheries, whether they are the commercial ones or the regular ones. The main difference is that the commercial red sex links have a lot of leghorn in them so they don't get as big. Thus they have a better feed to egg conversion rate. They don't have as big a body mass to support, so they don't eat as much to produce the eggs.
Some hatchery red sex links are made by crossing a Rhode Island Red rooster with a Rhode Island White hen. Some hatcheries use other breeds, but this is a common cross. Since those parent breeds both lay well, the red sex link hens lay well. But if you took a rooster from the same RIW flock and crossed it with a hen from the RIR flock, you would not get a sex link, but it would lay just as well. You just would not be able to tell which sex they were by down color at hatch. The important thing is that if the parent flocks lay well, the offspring will probably lay well.
Most chickens lay pretty well the first year they lay, but they start off with fairly small eggs. This protects the pullets because their body is not real mature. If they started out laying really big eggs, they would be more likely to have physical problems. The eggs will get bigger as they age and after their first adult molt, the eggs will get noticeably bigger. And after their first adult molt, they will continue to lay a lot of eggs. But after their second adult molt, the average hen will lay about 15% fewer eggs. They do not stop totally, but where you might have gotten 7 eggs each 8 days, you only get 6. And after each adult molt after that, the production drops more. For you and me, a couple of 15% drops probably won't be that noticeable, but if you have four hen houses with 5,000 laying hens in each, it becomes pretty noticeable. That's why the commercial operations replace the flock after a certain time. It's not that the flock stops laying totally, but the laying rate drops off enough that the efficiency of feed to eggs conversion rate is not there where they can make a profit. That's why the commercial laying breeds are designed to lay a whole lot of nice sized eggs for the first year but may suffer burn out. They are not selected for longevity of laying.
I know this is a long way around to answer your question, but if you select a pure breed from most hatcheries, you will get a chicken that lays pretty well. It will still drop off in production after the second and following adult molts. If you select a cross that comes from these breeds, they will be the same way. If you select a cross that is the commercial layers, they will also behave the same way, but you have a higher chance of them having burnout after the second molt. What I mean by burnout is that they are not designed for longevity, so they are more susceptible to physical problems as they age.