Using auto-fill shortcut when logged out doesn't load vault

### Steps To Reproduce

1. Logout.
2. Go to any website.
3. Use keyboard shortcut for "Auto-fill the last used login for the current website".
4. Login to BW.

### Expected Result

Logging in using the keyboard shortcut should be the same as clicking the BW icon.

### Actual Result

The vault will be and stay empty. "Last sync" is Never.
You have to sync the vault manually for it to load your login credentials.

### Screenshots or Videos

_No response_

### Additional Context

_No response_

### Operating System

Windows

### Operating System Version

_No response_

### Web Browser

Chrome

### Browser Version

_No response_

### Build Version

2023.12.1

### Issue Tracking Info

- [X] I understand that work is tracked outside of Github. A PR will be linked to this issue should one be opened to address it, but Bitwarden doesn't use fields like "assigned", "milestone", or "project" to track progress.

Start by tracing the browser extension's keyboard-shortcut path for auto-fill, then compare it with the BW icon flow after logout and login. Reproduce the issue in Chrome on Windows and verify that the shortcut loads the vault and updates the last-sync state without a manual sync.

Written by the indexing model from the issue text.