I did something similar, but used j-clips and basically built a rabbit hutch. No wood at all. It is sitting on a wooden frame. Your door is definitely better than mine.
According to one of the University websites (I can't remember which one), chicks raised on wire do not build up an immunity to cocci. They need to ingest a small amount of their poop that has seasoned a day or two so they can build up an immunity to whatever cocci is present in any of the chicks. The protazoa needs to develop a day or so after it is expelled from their system so it can become active when they re-ingest it.
You may be aware of all this so please bear with me. There are six or seven different types of protazoa that cause cocci in chickens. Developing an immunity to one does not give immunity against any of the others. Cocci can be a serious illness in chickens, even kill them. However, it takes a concentration to cause a problem. They can handle a small amount, although some protazoa are stronger than others. The protazoa can get concentrated in wet poop in the brooder or coop or run. If you keep your brooder dry, unless you get a really nasty strain, your chicks are unlikely to get sick from cocci. If they don't build up immunity, they can become very ill or die when they get to the coop.
What I'd suggest is having something that will hold a little of their poop and keep it dry, maybe a small tray with shavings. I use a piece of old window screening. You still have to clean it when it gets damp, but it is only a small area and you get most of the advantage of that droppings board and you can keep their water on the other side.
