But practically, as a person who runs sandboxed browser daily, it there is not "a great deal of convenience and restrictions". Even before sandbox, I'd download files to default location and later move some of them elsewhere -- so this is not really changing. A requirement to place files which need to be uploaded into a shared folder is somewhat annoying, but I found out that I don't upload that many files from browsers anyway.
Now you are moving goalposts. Initially, you were discussing restrictions on the file system.
I concur that the current desktop security model is probably unfixable. All the tools to improve on it are here though. Android kinda solved it by restricting every app to its own assets and files by default. If the app needs more, it has to ask for permissions.
The problem with user approval for every single action is decision fatigue. It is already happening on Android: every app is asking for a ton of permissions. And it turns out that many of them are not granular enough. For example, Signal needs the numbers from the address book to find contacts, but nothing else. If I want that feature, I still have to give Signal access to the entire address book.
But yes, it's true that issues with the cache can be resolved with this, but not saving outside of a single file path, and whatever utility that then moves the file outside of that single path would still need full permissions or be granted permissions.
It can be done, but at the cost of a great deal of convenience and restrictions.