I agree with all the reasons given to not mix those things. The will pick and choose what they want to eat, missing out on some other things. They will probably pick the corn out first, leaving feed behind so they are not getting the benefit of a balanced diet. They are usually pretty good about only eating what calcium they need, but there can be powder or fairly fine pieces of the oyster shell so you take that ability to regulate how much they eat away from them when you mix it. They can eat as much grit as they need if it is offered separately in its own container or if just tossed on the ground.
I don’t consider corn empty calories. If you look at the analysis of corn you’ll see it has a lot of good nutrients in it. There is a reason corn is a significant portion of a lot of animal feed for a lot of different animals. When I grew up on the farm our draft horses were fed corn to supplement their grazing in warmer weather and supplement hay in the winter to help maintain their fitness. When we milked the milk cow, we fed her corn to improve the quality and quantity of the milk she gave. In winter we fed corn to the chickens. Otherwise the chickens foraged for what they ate. We didn’t get much snow, they had access to the hay barn, they could scratch through horse and cow manure, and could find a lot of other forage even in our winters. I know that thought horrifies many people on this forum, but that’s the model small farms have used for their chicken flocks for thousands of years. Climate does play a part.
Many people fixate on the amount of proteins in a food when they are looking at the nutritional value of the feed. Because of the type of corn used and when and how it is harvested and stored, what we feed is usually around 10% protein. That’s low. You need enough of the other stuff to get that protein up to where it needs to be. That’s why corn or any other treat needs to be limited. It needs to form a small part of their overall daily diet. If you are feeding a commercial chicken feed, it contains a good balance of what they need. If you look at the analysis on the label you’ll see the only significant differences are protein and in the case of Layer, calcium.
No matter what treat you feed them, you are upsetting that balance. I’m all in favor of feeding them all kinds or “treats”: critters, grains, kitchen wastes, stuff from the garden, grass clippings, whatever, but I try to do that in some moderation. Don’t make any of this a significant portion of their overall food, especially day after day.
Yellow corn does add some color to the yolks. How much depends a lot on what variety of corn you use. They can get the carotenoids for that from various green leafy vegetables or wild plants. Certain veggies supply those. I notice a big difference in yolk color when I toss marigold blossoms, even dried dead brown ones, into the run. As Canoe said, it has nothing to do with nutritional value of the egg, but I just like the darker color.
There are a lot of good reasons to offer feed, grit, treats, and oyster shell separately. I can’t think of any good reason to mix them.