Firefox

Isn't your computer disk encrypted already?

Otherwise you seem to want jails or sandboxes to protect each app, with access denied by default. That sounds more like Android, or possibly Qubes OS: https://en.m.wikipedia.org/wiki/Qubes_OS

I'm a little confused as to how that would help with privacy/security.

When your browser is open and 'unlocked' a virus could still read the data.

It's the same thing with full disk encryption, if you get a virus on the running system it doesn't matter.

To protect it from ... what attack are you stopping here? If you don't know, and it sounds like you don't know, then forget it.

If someone roots your device, you still lose. If someone takes your device while you're browsing, you still lose. If your hard drive is unencrypted, you still lose.

They refuse to do it because the idea has absolutely no merit to it. If there's a virus on your computer that could steal your data, it can just wait till you unlock that data to steal it. There is zero practical benefit to implementing your suggestion.

"full-disk encryption" is the search keyword you're looking for

Sandboxie might help? (For Windows)

https://github.com/sandboxie-plus/Sandboxie