If I, as an application author, can only work on data in ways that are intended and officially blessed by another application, we basically have the situation we have on mobile where everything is siloed, and the state of the art is limited by the imagination individual application developers.
BTW both Safari and Firefox are not going to implement the File Access non-standards that Chrome pushes. They expose too much, there are no good ways to limit the exposure etc.
If they were like cars, only bulletproof armored personnel transports need apply.
The modern Internet is basically a failed state.
How about just subfolder access? Limit facebook.com's permissions to only the user/facebook/ folder.
When it asks the user for a file, can also specify the wanted access: read, write, read+seek, or read+write+seek. Requested format can also be specified, but the user should be allowed to ignore the requested format if wanted and instead specify an arbitrary file. For writes, estimated file size can also be specified as a hint, which can also be ignored. Then the user can type in a file name, or for the non-seeking modes, a pipe is also possible. For write non-seeking, the user can specify append or overwrite. For seekable files, a pipe is not valid. For writing to files, the user can also optionally specify the maximum size that the file is allowed to have.