The app "Documents" by Readdle work pretty much like you would expect from a Finder app in iOS.
Actually, since iOS 8 app extensions, any app that wish to offer that funtionnality can make itself available as an option in the iOS "File picker" (the menu that popup where you can pick files from iCloud Drive for example). You can add any app that support it to this menu (Document, Dropbox app, Google Drive app, etc.)
Documents work as a local and offline file system with folders and subfolders, file management features and can open a wide variety of filetypes (office, iWork, pdf, media, zip (and also unzip files)).
It's also possible to use it to store, for example, your Pages documents (instead of iCloud drive). From the file picker in Pages, you can select Documents as your source, browse it's filesystem and then pick a specific .pages file, open it, make changes to it, and close it. And then every changes will be saved directly inside the Document app, and not just as a "copy in Pages iCloud folder" like it was before. So it means you no longer have to create multiple copy of different files. (and all of that happen offline)
An app can create new files and move them to any folder in Documents too, and in Safari you can download directly to Documents, and even upload files from Documents to directly in Safari.
Basically, "Finder" for iOS.
Things have changed a lot since iOS 8.
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The biggest "drawback" is that for some reason, it's possible to sync a specific folder from a cloud service, like Dropbox or Google Drive to Documents, but that specific folders cannot be picked in another app like Pages to edit a file. But any "normal not a cloud sync" file or folder in Documents work as a Finder app would work.