Allow deep linking to default search engine dialogs

[emerick]

Resolves brave/brave-browser#42999

Makes the dialogs to set the regular default search engine and the default search engine for private windows navigable via deep linking. The link for setting the regular default search engine is brave://settings/search/defaultSearch and the link for setting the default search engine for private windows is brave://settings/search/privateSearch.

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:

• Clean profile
• Visit brave://settings/search/defaultSearch
• Ensure that default search dialog appears and works correctly
• Ensure that URL bar shows brave://settings/search/defaultSearch
• Ensure that accepting the new default search engine works
• After accepting (or cancelling the dialog) the browser should automatically navigate to brave://settings/search
• Test that the new default search engine works
• Perform all of the same testing for the private search (brave://settings/search/privateSearch)

[brave-builds]

A Storybook has been deployed to preview UI for the latest push

[simonhong]

👍🏼

[fallaciousreasoning]

nice work!

[brave-builds]

Released in v1.76.15

[LaurenWags]

Verified with

Brave | 1.76.23 Chromium: 132.0.6834.83 (Official Build) nightly (arm64)
-- | --
Revision | 8648b4550ee6f13d98522ea0437be7bbed94dba9
OS | macOS Version 15.3 (Build 24D5055b)

Using 1.74.48 Chromium: 132.0.6834.83 confirmed visiting brave://settings/search/defaultSearch and brave://settings/search/privateSearch redirected to brave://settings as the deep links don't exist in that version.

Verified test plan from #27223 (comment) for default search deep linking - PASSED

1. Clean profile 1.76.x
2. Close and relaunch to pull griffin
3. Visit brave://settings/search/defaultSearch
4. Ensure that default search dialog appears and works correctly

5. Select "Cancel" on the search dialog and confirm you are navigated to brave://settings/search

Screen.Recording.2025-01-20.at.11.13.52.AM.mov

6. Close this tab
7. Open a new tab and navigate to brave://settings/search/defaultSearch again and that the search dialog appears again
8. Ensure that URL bar shows brave://settings/search/defaultSearch
9. Ensure that you are able to change the default SE from this dialog

10. Confirm that after selecting "Set as Default" you are automatically navigated to brave://settings/search
11. Confirm that the newly selected default search engine works

Screen.Recording.2025-01-20.at.11.15.34.AM.mov

12. Close existing settings tab
13. Open a new tab and navigate to brave://settings
14. Use side navigation to get to Search section
15. Click on "Change" button next to the default SE
16. Confirm URL reads brave://settings/search/defaultSearch
17. Cancel this dialog, confirm you are navigated to brave://settings/search
18. Open a new tab and navigate to any site
19. Highlight some text on a page and right click
20. Confirm the context menu uses your chosen SE from previous step

21. Close and reopen browser
22. Open a new tab
23. Confirm previously selected SE is retained for searching from URL bar

Example	Example	Example

Verified test plan from #27223 (comment) for private search deep linking - PASSED

1. Clean profile 1.76.x
2. Close and relaunch to pull griffin
3. Visit brave://settings/search/privateSearch
4. Ensure that default search dialog appears and works correctly

5. Select "Cancel" on the search dialog and confirm you are navigated to brave://settings/search

Screen.Recording.2025-01-20.at.11.40.42.AM.mov

6. Close this tab
7. Open a new tab and navigate to brave://settings/search/privateSearch again and that the search dialog appears again
8. Ensure that URL bar shows brave://settings/search/privateSearch
9. Ensure that you are able to change the default SE from this dialog

10. Confirm that after selecting "Set as Default" you are automatically navigated to brave://settings/search
11. Confirm that the newly selected private search engine works in private window

Screen.Recording.2025-01-20.at.11.42.48.AM.mov

12. Close private window and the existing settings tab in normal window
13. Open a new tab and navigate to brave://settings
14. Use side navigation to get to Search section
15. Click on "Change" button next to the private SE
16. Confirm URL reads brave://settings/search/privateSearch
17. Cancel this dialog, confirm you are navigated to brave://settings/search
18. Open a new private window and navigate to any site
19. Highlight some text on a page and right click
20. Confirm the context menu uses your chosen SE from previous step

21. Close and reopen browser
22. Open a new private window
23. Confirm previously selected SE is retained for searching from URL bar

Example	Example	Example

24. Confirm default SE is unaffected in normal window (retains default selection)

Screen.Recording.2025-01-20.at.11.53.46.AM.mov