Remember the old television commercial jingle "brown eggs are local eggs, and local eggs are fresh"? That's because brown eggs come from chickens who can best tolerate our colder winter weather.
Regarding the richness of the yolks, brown eggs truly do not have richer yolks than white eggs. The difference in color comes from pigment, and it doesn't affect the yolks. What does in fact affect the flavor or richness of the yolks (and the darker color of the yolks) is the diet of the hens who lay the eggs. Often, better feed that includes fresh vegetables results in richer egg yolk flavor. This is considered a good thing by most people, and it is a big plus for those who use the eggs for baking. If you find the yolk flavor to be too rich or otherwise unappetizing, you should look at methods of diet control rather than restrict yourself to white egg layers.
That said, it's true that the white egg layers tend to be more high strung and less likely to be fun for your kids. The white egg layers will also have a hard time during our harsh winters. Two of the most affectionate, friendy and calm breeds are the Buff Orpington and the Speckled Sussex. Both are kid-friendly, pretty, good layers, and both lay very light brown eggs (almost cream colored)---so the color would be close enough to white but again, yolk flavor is about diet, not the color of the egg shell. I would heartily recommend these two breeds for your family.
To make a milder egg yolk flavor, feed your chickens blander foods---avoid strong tasting vegetables and stick with greens, watermelon, etc. Commercial layer pellet feed is an ideal base feed---bland but relatively complete, nutritionally. Do some research on the topic of diet affecting egg flavor for more information. But again, I cannot recommend the Buff Orpington and Speckled Sussex enough! ALSO, "Easter Eggers" (called EEs, they are also sold as Araucanas or Ameraucanas at feed stores), lay pretty blue, blue/green and green eggs that often delight kids. Here again, the color of the egg shell doesn't affect the taste of the egg---they are simply pretty eggs to look at in a basket...plus EEs are friendly birds, very pretty, cold hardy and super layers. I love EEs!