From what I gather from the docs [1], this API gives you a FileSystemDirectoryHandle object, and then you just call getDirectoryHandle() on that to recursively read the the entire filesystem. The spec [2] has some vague suggestions about blacklisting certain particularly sensitive files, which doesn't seem reassuring.
[1] https://developer.chrome.com/docs/capabilities/web-apis/file...
[2] https://wicg.github.io/file-system-access/#privacy-wide-acce...