There is quite a controversial opinion on this topic.
Some say absolutely nothing should be fed to a chicken other than a complete and balanced diet. I agree with this, for the most part. As you know, anything bagged, dried, or sold as a chicken treat is no good. Its likely high in fat, with very little nutritional value. Causes fatty liver disease etc, you already know all of that.
But, I really think that greens, spouts, and anything green is very beneficial, and important in a chickens diet.
Chickens eat a diet mostly of grains. They're built for that, but it also means its low in antioxidants, and omega 3's are always helpful since grains are very very high in omega 6's, which create inflammation. Another touch on this, big brand chicken feeds can contain waste kitchen products. So basically whatever is leftover from grain based people food. This can be bread, baked goods, etc. That is going to be very high in omega 6's. There are lots of plants, such as hemp seeds, that have omega 3's. Antioxidants are very important since we all have free radicals, which creates oxidative stress, premature aging, cancer, and lots of other diseases. If you don't get antioxidants, you will succumb to those side affects. Thats why chickens should get lots of forage, fruits, and plants, especially laying hens who age so quickly.
For these reasons above, I think that greenery, forage, leafy greens, and non starchy vegetables and berries are incredibly important for chickens.
I think a fodder system is a great idea! Feeding additional things to chickens can be tricky. The problem with anything sold as a treat for chickens is that it has little to no nutritional value. There for it is only diluting the nutrients in a feed. When your taking leafy greens, forage, fodder, and especially micro greens, those foods are very high in nutrients, and very bioavailable as well. With that in mind, greens/forage shouldn't be diluting the diet to the point where it is causing harm. I personally think that it is doing so so much more good than harm.
As you know, my opinion is likely going to be different than others opinions. I do a lot of research on natural nutrition for my dogs, and a lot of it is linked since you feed dogs chicken. You are what you eat when it comes to nutrition, so a poorly fed chicken that you feed to a dog is going to be a poorly fed dog. Because of that I do a lot of research regarding poultry nutrition as well. I'm fairly confident in my opinion.
Oregano is a fantastic immune booster, along with Echinacea, lemon balm, high quality probiotics, ginger, and most all other herbs.