I'm genuinely not all that familiar with what PWAs can and can't do... Would a, for example, Word-like PWA be able to both load and save files to disk?
Just like any web app, it would be able to read specific files of a user's choosing using an HTML "file" input, and it would be able to save files by initiating a download. The app would not be able to provide its own UI for selecting the file to open/save but rather it would have to rely on the browser's UI for doing those things.
As an example, if I were using a PWA-based schematic editor, I couldn't just hit Cmd-S and save it back to disk without having to choose the download location etc every time?
You would have to do that, unless you are willing to only save the data in the app's storage itself while you're working on it. Similar to the model typically used with mobile apps, for example.