I get itchy legs in the winter time and it is a skin issue like you are having, i think. I also get some food related skin trouble occasionally that is just like what you are having. My first impression, since you have eliminated all topical factors, is that it is food related and not topical at all. Simply it is some kind of autoimmune response, where your body is attacking itself for something that it thinks is a problem, but isn't a problem.
You might want to start an elimination diet program to see if you can identify the type of foods that might be causing the problem. It is actually quite challenging to do, but you have to start eliminating stuff from your diet for like a month to see if something is causing the problem. Most of the time, I see dietitians suggest eliminating gluten, dairy, eggs, soy, fast food, and alcohol. Then, you can add one item back after the 30 days and add only that item back. Just see how it is after a couple of days and see if you feel different.
Here is a pretty good explanation: https://draxe.com/elimination-diet/
We use a soap called Magic Soap, which is made from organic ingredients, no chemicals, and includes Tea Tree Oil. You can use it for pretty much anything you would need soap for, in different degrees of concentration. You can use it as regular soap, after you dilute it a lot. Please follow the directions carefully because it is concentrated and will cause eye and skin irritation if you use it straight from the bottle.
They also have a bar version of the soap, which might be easier to deal with.