“Worry is interest paid before it is due”. I stole that phrase from someone else on here. I know it is hard to do but try to find out if you have a problem before you worry too much. Reality is often much nicer than our imaginations though sometimes reality sucks.
Your problem with Cocci is not so much that you have it but your ground is wet. The problem with cocci is not that it is in the chicken’s digestive system but that the numbers get out of hand. That cocci bug thrives in wet soil that has manure in it. The chickens peck in that and eat a lot of the ”eggs” which then develop inside the chick and the numbers quickly get out of hand. There are a couple of strains of cocci that are really damaging because of which part of the digestive tract they attack, but the majority of the problems people have with cocci is when wet is involved, either dirty water, a wet brooder, or wet ground.
Once cocci is in your flock, cocci is always in your flock. Even if you were somehow able to eliminate it from your chickens and from that patch of ground, it will just come back in from outside the sterilized area. You have it but you can deal with it.
Chickens will develop immunity to a certain strain of cocci with two to three weeks exposure to it. The secret is to allow them to develop that immunity without the numbers getting out of hand and getting them sick. I don’t know how that immunity works but it does.
The medicated feed probably contain Amprolium. I’d check the label for active ingredients before I depended on it, but the dosage of Amprolium in medicated feed is meant to be a preventative, not a cure. It reduces the ability of the cocci bug to reproduce so it limits how many “eggs” are laid but it still allows some reproduction so they can develop immunity. It is still possible for them to get sick and even die from cocci while they are on medicated feed, but it does help reduce the risk. You still have to know the symptoms and be ready to treat. And try to keep areas dry.
What is happening with yours is that the adults have the bug in their system and they are producing those “eggs”. Those eggs get in your wet area and thrive. Even if your chicks are on medicated feed and not producing many of those eggs themselves, they can easily become fatally sick because of the cocci bug thriving in that wet soil.
So how do you manage? If you can, dry that area up. Trust me, I know how hard that can be in some circumstances. It may be impossible. Still, if you can, that should be your main effort. Maybe put pallets or something over it to try to keep them out of the worst of it.
It sounds like you may be planning to use a broody instead of an incubator and brooding them yourself. If you are incubating and brooding yourself, keep the brooder pretty dry, change water completely every day if they can poop in it or scratch bedding in it at all, and feed them small amounts of dirt from the run to expose them to the bug so they can work on their immunity. One fairly common happening on this forum is that someone broods the chicks in isolation, feeds them medicated feed, but does not introduce the cocci bug to them. Then when they are old enough they stop feeding medicated feed, put them on the ground where they are introduced to the bug, and the chicks get sick and maybe die. Your goal is to strengthen their immune system, not raise them in a sterile environment.
If they are being raised by a broody you can just let them go and see if you have a problem, being ready to treat at the first sign of a problem. If she has a drier place to hang with her chicks other than that mud hole it might work, but they are not going to want to be on that concrete much. She’ll want them somewhere they can scratch and peck, even if it is mud. Still, this may be your best option.
I personally like a broody to raise her chicks with the flock from Day 1, but we have to have some flexibility and be ready to adapt. If you can isolate her and the chicks from that wet area for about three weeks they should develop the immunity they need. They will be exposed to the bug with the broody hen pooping in that area. After three weeks introduce them to the flock before she weans them so she can handle integration for them.
No matter what you do, watch them closely and be ready to treat at the first sign of cocci.
Good luck with it. Many of us have somewhat similar situations. My run can be really wet when the weather sets in wet for a while. I’ve only had one outbreak of cocci, that was with a broody hen raising them in really wet weather, but my broodies have a somewhat drier area to go and the manure load is not that great in that dry area.