I’ve worked on the Native File System API on Chrome.
It’s available through Origin Trial at the moment.
That said, Web apps can currently emulate reading files and writing to disk.
Take a look at this library, which also works as a poly fill:
The vast majority of the work I do can be done off-line with files stored on disk, minus stuff that obviously needs connectivity (e.g. Slack). Writing text, writing code, reading/editing documents and spreadsheets, doing CAD (mechanical and electronics), etc. During the week I live in the city, but on weekends and occasionally for longer stretches of time I like to head out to our rural house for peace and quiet (and focused work). Over time, the number of applications that ought to work off-line has been slowly shrinking, whether it's because they insist on using cloud storage instead of local storage, or licensing checks, or whatever. It's really really disappointing, and I would personally be delighted if I could get work done without an Internet connection.
Obviously this is not a great workflow for certain things like system administration tools that are specifically designed to work with the local machine's filesystem, but that's not really a good use case for PWAs in the first place.
Forcing user to generate/download ZIP at some well-defined points in time is a horrible UX idea; in addition it leads to desynced content of files and local storage. Browser's local storage can also disappear if SQLite gets corrupted, making it a nice SPOF for all PWAs that are installed. Imagine not losing data just for one app, but for all of them.