The deep litter method is intended for inside the coop where it stays dry. The risk to putting shavings, straw, or any other organic matter is that it can get moldy if it gets wet. Mold can be harmful to chickens. Some people have developed strategies to deal with it or otherwise have been lucky. Just because something can happen does not mean it will happen each and every time like clockwork, but the possibility for a big problem is there with wet moldy organic matter.
I don't know how big your run actually is, but if you have 5 chickens in there and it still has some grass, it sound like it is fairly large. I don't think you have to worry about putting anything in there at all. I find that the more space I give the chickens, the less I have to manage the poop. Some people build the bare minimum size they can get by with and have to rake or remove the poop every day, especially when they live in suburbia where space and smell are real important. I'm not criticizing them in any way. They are doing the best they can in their circumstances and most do a real good job. You deal with the circumstances you have.
Some of us have much larger runs and never have to worry about managing the poop. It does not build up to where it causes a problem. If you still have grass in yours after a summer, your run should fall in this category. If it does not smell when fairly dry, you are doing OK.
Unless you have a small covered run, it is practically impossible to keep any run dry when it rains. You can try to divert water away and such to try to keep most of it out, and you can maybe do some things to help it drain faster, but a run will get muddy, especially where they like to take dust baths. A wet run can smell a bit. How bad it smells depends on how thick the poop is and how long it is wet. My chickens don't seem to mind a wet muddy run. They still get out there and play around, but occasionally I'll see one that gets something stuck on her foot trying to clean it off.
A run that is normally wet is unhealthy, but one that is occasionally wet is OK. It takes some time for the numbers of the bacteria or protozoa to build up to where it causes a problem. I can't give any rules to help you judge that. What sticks out in my mind is that you still have grass after a summer. That tells me that you have a big run and it was not too wet, since wet ground makes it even easier for them to destroy the vegetation. I just don't think you have a problem.
I'll give a link to Pat's Muddy Run page. It may help you.
Pats Big Ol' Mud Page (fixing muddy runs):
https://www.backyardchickens.com/web/viewblog.php?id=1642-fix-a-muddy-run
As far as nutrition, commercial feed has everything in it the hens need to stay healthy and produce plenty of decent sized eggs. It is specially formulated to take care of all their needs. You do not need anything else for a balanced diet. Since you don't free range, you have the risk that if you feed too much of anything else you can upset the balanced diet provided by that complete well balanced feed. The general rule is to only feed enough of something else that they can clean it up in 10 to 20 minutes. I can assure you, many of us violate that and don't suffer huge consequences. But the key to that is to not feed the same treat every day. Give them a variety of treats but don't overdo it much. Think of it this way. Tomatoes are great for you nutritionally. But if tomatoes make up a big portion of your diet, you are probably not getting all the vitamins, minerals, protein, fats, fiber, amino acids, and whatever else you need for a balanced diet. Why would it be any different for chickens? When they are out free ranging, they eat grass and weeds, grass and weed seeds, creepy crawlies, flying and hopping things, and a lot of other disgusting things we won't mention. They don't just eat grass seed and nothing else but balance their diet. Moderation in any one thing is the key.
I do think the eggs are better and have more character if they get some fresh stuff to eat. Tests have shown that free ranging chickens produce eggs with more of the trace elements and such that we need. But you can't free range yours because of the road. Since you still have grass and weeds in your run, they are getting stuff form that, but a variety of additional treats like kitchen scraps, grass clippings, stuff from you garden if you have one, especially things like cabbage or broccoli leaves, excess tomatoes or squash, whatever, are great and will help your eggs. But if all you have to offer is commercial feed, the nutrition is sufficient for them to have a healthy productive life.
Good luck! Sounds like your girls are living a good life.
