Fix #8242: Fix bug in menu seeing private browsing as false

[Brandon-T]

Summary of Changes

• The publisher for isPrivateBrowsing does not update the value immediately before being triggered, so it causes a bug sometimes where the menu isn't updated correctly

This pull request fixes #8242

Submitter Checklist:

• Unit Tests are updated to cover new or changed functionality
• User-facing strings use NSLocalizableString()
• New or updated UI has been tested across:
  • Light & dark mode
  • Different size classes (iPhone, landscape, iPad)
  • Different dynamic type sizes

Screenshots:

Reviewer Checklist:

• Issues include necessary QA labels:
  • QA/(Yes|No)
  • bug / enhancement
• Necessary security reviews have taken place.
• Adequate unit test coverage exists to prevent regressions.
• Adequate test plan exists for QA to validate (if applicable).
• Issue and pull request is assigned to a milestone (should happen at merge time).