Why Don't Browsers Let Web-Apps Write To The Local File System?

I mean, I know why. It’s a security thing.

But why couldn’t a browser have an API for scripts to read / write the file system and a security feature where the web-app has to ask and be given permission by the user before it runs? (Just as Android apps. have to tell you what permissions they need before you install them.) Couldn’t the browsers successfully police this?

Surely if the browser manufacturers were to offer this capability, they’d more or less kill native Windows / Macintosh application development overnight and become the default platform for desktop computers. (So maybe Microsoft don’t have the incentive, but Google and Firefox do.)