An update after about a week.
If I'm using regular shampoo instead of my medicated shampoo for more than a couple days, it gets very flaky, cracked and even starts bleeding. I've always thought this was due to the scalp condition. So, I went into this expecting things to be hell by the start of the weekend.
Not so. My scalp is actually perfectly fine, no issues. To me, that proves the hypothesis that my scalp has been reacting to regular shampoo, rather than having a condition that requires treatment.
This is consistent with what happened when I stopped using soap on my face many years ago. I used to get these red, sensitive blotches on my face, and tried a lot of different approaches to dealing with them until I just stopped using soap all together. Then, they went away and my skin is fine.
I still use body wash for my nether regions, and have continued to do so during this experiment. I have limits.
The hair is okay, but greasy and a bit hard to manage. I have very, very greasy hair naturally, so I expected this. I have taken to using very hot water at the end of the shower to help wash some of it out, then toweling it down thoroughly to get some of the excess out. It's not horrible, but feels a bit heavier and has a flat appearance to it. On balance, I'd rather use my (expensive) medicated shampoo and have hair that feels and looks better than it does now. It's leaving me a little self-conscious.
I see people saying to give it a couple weeks to my body can adjust and self regulate a bit more, so I'm going to do that and see how it goes. I'll make a call next Sunday night whether it's improving at all or whether to go back.
All in all, it's been much more successful than I thought. My experience has me convinced we're basically being sold soap to wash out our body's natural oils, and then conditioner to put some oils back. For most people, going without either will work just fine.