With chicken genetics you are talking about a gene pair at each location on the chromosome. A rooster gives one copy of a gene at each location on the chromosome to all his offspring, male and female. If both genes at that location are the same, the offspring gets the same thing. But if there is a difference in the genes at that gene pair, then he randomly gives one of those two genes to all his offspring. Since a rooster gives one copy at each location, he contributes to all the characteristics of his offspring, including egg laying ability.
A hen does not do that. She gives one copy from each location on the chromosome to her sons, but she does not give any sex linked genes to her daughters. That means the hen decides which of her offspring are male and which are female. That means she does not contribute as much of the genetics to her daughters as the rooster does.
As Mrs. K said, there are many different gene pairs that contribute to egg laying traits. I don’t know how many of those are sex linked but it stands to reason some are. As a minimum a rooster will contribute just as much as the hen to the genetics of egg laying.
A rooster does not lay an egg. You can’t tell by looking at the rooster what genetics he will contribute, but if his mother and his grandmothers were good egg layers, then the rooster probably passes on some pretty good genetics. You can see what the hen is leaving in the nest so you have a much better idea of what genetics she is contributing. The traditional way and about the only practical way to improve egg laying quality is to hatch your best eggs and keep your breeding chickens from them.
Most hatcheries use the pen breeding method as a way to maintain the genetic diversity of their flock. That means the genes are going to be mixed up and not as consistent as breeders trying to eliminate certain genetic diversity from their flock. If you are wanting to consistently get larger eggs from your flock, then that means you are reducing genetic diversity for that trait. A lot of that success depends on the person selecting which chickens get to breed too. If they select for certain traits and know what they are doing, either in a hatchery or a private breeder, then those traits will be enhanced. But if they are no selecting for a specific trait, then that trait could be all over the place.
If you can find a breeder that is breeding for the trait you want, then you are way better off with their chickens. But if you find a breeder that is breeding purely for show and only breeding for the traits the judge sees, you may get a chicken with perfect SOP eye color but you don’t know anything about the egg laying traits the hens might have. The judge does not see the hen’s eggs.
You can improve the egg laying ability of your flock by hatching your best eggs and selecting hens and roosters to breed from those, always going with your best egg laying hens. If you can get your starting stock from a breeder that is actually breeding for your goals, then you are starting off in a real good place.