[iOS] - Fix private-mode/normal-mode flip when opening tabs

[Brandon-T]

Security Review

• https://github.com/brave/reviews/issues/1796

Summary

• Fix switching modes programatically
• Fix the user being stuck in a mode where the tab-bar icon won't work!

Resolves https://github.com/brave/internal/issues/1232
Resolves brave/brave-browser#42357

Submitter Checklist:

• I confirm that no security/privacy review is needed and no other type of reviews are needed, or that I have requested them
• There is a ticket for my issue
• Used Github auto-closing keywords in the PR description above
• Wrote a good PR/commit description
• Squashed any review feedback or "fixup" commits before merge, so that history is a record of what happened in the repo, not your PR
• Added appropriate labels (QA/Yes or QA/No; release-notes/include or release-notes/exclude; OS/...) to the associated issue
• Checked the PR locally:
  • npm run test -- brave_browser_tests, npm run test -- brave_unit_tests wiki
  • npm run presubmit wiki, npm run gn_check, npm run tslint
• Ran git rebase master (if needed)

Reviewer Checklist:

• A security review is not needed, or a link to one is included in the PR description
• New files have MPL-2.0 license header
• Adequate test coverage exists to prevent regressions
• Major classes, functions and non-trivial code blocks are well-commented
• Changes in component dependencies are properly reflected in gn
• Code follows the style guide
• Test plan is specified in PR before merging

After-merge Checklist:

• The associated issue milestone is set to the smallest version that the
  changes has landed on
• All relevant documentation has been updated, for instance:
  • https://github.com/brave/brave-browser/wiki/Deviations-from-Chromium-(features-we-disable-or-remove)
  • https://github.com/brave/brave-browser/wiki/Proxy-redirected-URLs
  • https://github.com/brave/brave-browser/wiki/Fingerprinting-Protections
  • https://github.com/brave/brave-browser/wiki/Brave%E2%80%99s-Use-of-Referral-Codes
  • https://github.com/brave/brave-browser/wiki/Web-Compatibility-Exceptions-in-Brave
  • https://github.com/brave/brave-browser/wiki/QA-Guide
  • https://github.com/brave/brave-browser/wiki/P3A

Test Plan:

[brave-builds]

Released in v1.75.34

[kjozwiak]

Verification PASSED on iPhone 11 running iOS 18.1 using the following build(s):

Brave | 1.75.34 Chromium: 131.0.6778.69 (Official Build) nightly (64-bit) 
--- | ---
Revision | 8874653c268c...
OS | iOS

Using the STR/Cases outlined via brave/brave-browser#42357 (comment), reproduced the original issue with 1.75.32 where clicking on media via Playlist and selecting Open In New Private Tab opens the media in a regular tab rather than a private tab.

Using the same STR/Cases mentioned above, verified that the above was working as expected with 1.75.34 Chromium: 131.0.6778.69.