“Take Control of iCloud” does not seem to provide a suggestion to my question.
Is there a way to get rid of the folder systematics and organize things the way I need it? Just as the Finder on my Mac lets me do it. How am I supposed to find a certain file without opening several folders where it might be hiding when its format is one of many possible ones?
You can create your own folders, of course, but I don’t think you can avoid the app-specific folders.
The problem I’ve been having is that files I add or modify on one device aren’t always updating quickly on others, and sometimes they don’t update at all (there’s a little cloud icon next to them in the Finder view). With one set of files, I had to move them over manually via AirDrop because nothing I could do would convince them to copy to iCloud. And there was nothing special or unusual about them—other similar files in similar folders were working fine.
In Germany we would say “Das ist Mist!” (Oh crap!)
Mother Apple knows what we need?
Maybe she does. But in the end she decides what is good for us and the faithful flock will follow. 
You can create your own folders, of course, but I don’t think you can avoid the app-specific folders.
True, but you can just ignore the app specific ones and just work from your own folders. I have a big subject based folder structure with multiple file types in them.
That would still leave you with the hassle of moving a Pages doc to another folder. Or - even worse - a PDF which could have been created or altered by more than one app.
Yes not ideal or like on a computer, but I don’t have to go hunting for my files on a particular subject in the app specific folders like your opening post. I can put them where I want in my own subject based folder structure, and they stay there through edits.