(I don't totally understand the difference between a Chrome app and an extension that turns itself into a PWA, both are browser-specific and both are distributed through the Chrome web store, right?)
Ideally, a directory for access/storage enough to use SQLite as via WebAssembly should be available to plugins with a request/approval for read (and optionally write) access ... to me, that should be a goal of any file system interface presented via WebAssembly, is the option for something that at least supports that use case.
But yeah it seems network sockets are going away. Honestly that seems good from a security perspective but I will miss the mosh app for CrOS.
For testing, security research and so on you can always modify the browser config or use a proxy.