Are you sure this is still true? I thought nowadays it's mostly what supplements they put in the feed. These supplements are sold exactly with this kind of promise.
"Handbook on Natural Pigments in Food and Beverages"
Chapter 14 - Feed Additives for Influencing Chicken Meat and Egg Yolk Color
Pigmentation of egg yolks and poultry tissues (mainly skin and fat) directly reflects the contents of carotenoids in the feed of birds. In most countries of the world, consumers prefer pigmented egg yolks, whereas pigmented poultry tissues are less desired. Especially in the southern part of Europe, eggs with a golden-orange tone of yolks are preferred. This is achieved by supplementing feeds of birds with both yellow and red carotenoids. For this purpose, nine carotenoids are approved as feed additives in the European Union, five natural and four artificial products, six with a yellow color and three with a red color. This chapter describes the ways of pigmenting egg yolks and poultry tissues, how to measure the achieved color, the factors affecting the color of the product, the deposition rates of carotenoids, and the contribution of carotenoids in egg yolks to humane intake.
Poultry cannot produce carotenoids on their own, which is why the color of egg yolks depends on the carotenoids combination in their diets. However, it is not only the quantity of pigments that matters but also their bioavailability and color intensity. With Lucantin® Yellow (C-30 ester) any egg yolk color between light pastel yellow and reddish is possible.
There is one brand of eggs, happy egg co heritage breed that has some of the deepest color yolks I've seen with a different smell and taste. Significantly tastier. I've tried a lot of brands and price categories and haven't found ones like those.
Of all of the farm products discussed in the thread, eggs show the biggest difference in my opinion