[Android] fix update search engine text on brave widget

[tapanmodh]

Resolves brave/brave-browser#27749

Submitter Checklist:

• I confirm that no security/privacy review is needed, or that I have requested one
• 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 lint, 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/Custom-Headers
  • 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:

1. Open app -> Open 3-dot menu -> settings -> Home screen widget
2. Tap Add to add widget to device home screen
3. Background brave app, confirm placeholder text on brave widget references current SE
4. Open settings -> Search engines -> Standard Tab -> change SE
5. Background brave app again, check placeholder text on brave widget

[wchen342]

LGTM

[deeppandya]

LGTM

[kjozwiak]

Reproduced the original issue on Pixel 6 running Android 13 using the STR/Cases outlined via brave/brave-browser#27749 (comment) with the following build(s):

Brave | 1.49.41 Chromium: 109.0.5414.87 (Official Build) canary (32-bit)
--- | ---
Revision | 2dc18eb511c56e012081b4abc9e38c81c885f7d4-refs/branch-heads/5414@{#1241}
OS | Android 13; Build/TQ1A.230105.002

screen-20230120-003317.mp4

Verification PASSED on Pixel 6 running Android 13 using the following build(s):

Brave | 1.49.45 Chromium: 109.0.5414.87 (Official Build) canary (32-bit)
--- | ---
Revision | 2dc18eb511c56e012081b4abc9e38c81c885f7d4-refs/branch-heads/5414@{#1241}
OS | Android 13; Build/TQ1A.230105.002

Using the STR/Cases outlined via brave/brave-browser#27749 (comment), ensured that the string/text under the Home Widget is dynamically being updated when changing the default SE within Brave as per the following:

Example video of the the issue being fixed --> https://youtu.be/gZY2wASSj6Y