I've been thinking about this for a while. We have the "camera roll" in the photos app, and it's accessible to all apps that can take or use photos right? Add a lot more of these storage silos. One for text/word processing files, one for music, one for spreadsheets, etc. Devs could perhaps request new ones from apple.
That replaces the bits of a file system you really need, you can share documents between apps easily, and you don't get the complexity and security issues you get with file system access.
Last bit of the puzzle:
In safari, we can now have an 'upload' button. Hit it, it asks first what kind of document you want, then presents the data (search at the top, then a list of documents by document name rather than file name). we can finally upload photos to websites.
In email, there's an attachments button. Same thing happens, it asks what kind of document then shows the list, but you can add multiple documents with a tick then hit 'done'.
I reckon that would tie up quite a few issues people have with ios, and pretty cleanly & apple-like too.
