For the eggshells, eggshells alone may not provide enough calcium (there is some in layer feed, but needs can vary greatly bird to bird and season to season). I mix my saved shells 50/50 with crushed oyster shell and offer them all the time in separate feeders, I have 40 birds, so I have several in various places. That way it is available all the time and those that need it can take it when they want.
Feed is a complicated discussion. This post in this thread, see post #4, does a good job in short space on the fat question:
https://www.backyardchickens.com/threads/how-much-fat-is-too-much-in-layer-ration.1648803/
My personal experience is that excess carbs have been the worst cause in a large number of cases. A lot of corn, scratch grains, and sunflower seeds can do it. And "a lot" is actually very little, they are small. Greens generally don't contribute to fat, but too much of anything upsets a delicate balance and can cause problems. And some greens can affect calcium absorption. Any time they fill up on one thing, they don't eat their feed. I did see on your early post they list ingredients from sunflowers, so I assume they mean seeds, since they do not specify. When you have larger bodied birds that are that way naturally, weight is a particular concern. Some other breeds are more naturally lean, and they also seem to be the same birds that are more active. Which came first..... I don't know. And genetically prone birds are just going to more easily pack the fat on. That is why so many posts say over and over, don't give any treats that make up more than 10% of the total diet. And that is why whole grain feeds do not work for all birds. Some birds are fine, some birds will only pick out certain parts they like. I would not worry about insects they find free ranging, those are a natural diet and should be fine. I have always free fed, they take what they need, and if treats are kept to a bare minimum, I've never had a problem. Mine are outside all day and finding whatever bugs are out there, along with occasional frogs, lizards and mice. With the birds I have had that got fat, I don't hatch from them so as to not pass that particular trait on to chicks. Honestly, those ones have tended to have shorter lives. My batch of black sex links I think there were only 2 that made it to 3 years old. There are always exceptions, big birds can live a long time to. There are SO many variables. And there isn't anything you can do about genetics. I have changed feed brands before, when my birds were having issues, and sometimes the feed did make a difference. Same feed others use and are happy with. Different birds. No flock is exactly the same as another, again, too many variables. Don't be afraid to try different brands, or even rotate brands or formulas a little so they get a little variety over time. There is no one right answer, it's what you have available and what works for your birds. I feed an all flock or grower feed instead of layer, in the 18-20% protein range. I have mixed ages (some quite old) and roosters, so I don't want them all being forced to consume the calcium in layer feed. I have almost no shell quality issues since they have access to the separate calcium.
Feed is a complicated discussion. This post in this thread, see post #4, does a good job in short space on the fat question:
https://www.backyardchickens.com/threads/how-much-fat-is-too-much-in-layer-ration.1648803/
My personal experience is that excess carbs have been the worst cause in a large number of cases. A lot of corn, scratch grains, and sunflower seeds can do it. And "a lot" is actually very little, they are small. Greens generally don't contribute to fat, but too much of anything upsets a delicate balance and can cause problems. And some greens can affect calcium absorption. Any time they fill up on one thing, they don't eat their feed. I did see on your early post they list ingredients from sunflowers, so I assume they mean seeds, since they do not specify. When you have larger bodied birds that are that way naturally, weight is a particular concern. Some other breeds are more naturally lean, and they also seem to be the same birds that are more active. Which came first..... I don't know. And genetically prone birds are just going to more easily pack the fat on. That is why so many posts say over and over, don't give any treats that make up more than 10% of the total diet. And that is why whole grain feeds do not work for all birds. Some birds are fine, some birds will only pick out certain parts they like. I would not worry about insects they find free ranging, those are a natural diet and should be fine. I have always free fed, they take what they need, and if treats are kept to a bare minimum, I've never had a problem. Mine are outside all day and finding whatever bugs are out there, along with occasional frogs, lizards and mice. With the birds I have had that got fat, I don't hatch from them so as to not pass that particular trait on to chicks. Honestly, those ones have tended to have shorter lives. My batch of black sex links I think there were only 2 that made it to 3 years old. There are always exceptions, big birds can live a long time to. There are SO many variables. And there isn't anything you can do about genetics. I have changed feed brands before, when my birds were having issues, and sometimes the feed did make a difference. Same feed others use and are happy with. Different birds. No flock is exactly the same as another, again, too many variables. Don't be afraid to try different brands, or even rotate brands or formulas a little so they get a little variety over time. There is no one right answer, it's what you have available and what works for your birds. I feed an all flock or grower feed instead of layer, in the 18-20% protein range. I have mixed ages (some quite old) and roosters, so I don't want them all being forced to consume the calcium in layer feed. I have almost no shell quality issues since they have access to the separate calcium.