I found the reason. It turned out that despite uninstalling OneDrive, there were registry entries pointing to the \Users\Name\OneDrive\Documents (and desktop) folder, not Users\name\Documents. After changing the registry, everything is fine.
@leocg
It is my search engine in section Others. If I add another one now, I can edit or remove it, but what I added earlier is like in the screenshot - Edit option is gray and no Remove.