`unread-anywhere` should work after the page unloads

## Action

1.  Click button <img width="53" height="41" alt="Image" src="https://github.com/user-attachments/assets/8dc5094a-22f0-480e-a6b1-0be43953c16f" />
4. Close tab before it's done

## Expected

3. The new notifications should open in new tabs

## Actual

3. Nothing happens

## Fix

Move the API call and tab opening to the background.ts. The content script just wraps the message + UI part