Option for Index other search engine

[deeppandya]

Resolves brave/brave-browser#38129

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:

Index other search engines option inside Search engines should be disabled by default to match with desktop.

[mkarolin]

strings++

[tapanmodh]

lgtm

[bsclifton]

++

[bsclifton]

Looks like we have a presubmit error w/ gn check.

ERROR at //brave/browser/search_engines/private_window_search_engine_provider_service_android.cc:11:11: Include not allowed.
#include "components/search_engines/search_engines_pref_names.h"
          ^----------------------------------------------------
  It is not in any dependency of
  //chrome/browser:browser
  The include file is in the target(s):
  //brave/browser/android/preferences:java_pref_names_srcjar
  which should somehow be reachable.
___________________
ERROR at //brave/browser/search_engines/normal_window_search_engine_provider_service_android.cc:14:11: Include not allowed.
#include "components/search_engines/search_engines_pref_names.h"
          ^----------------------------------------------------
  It is not in any dependency of
  //chrome/browser:browser
  The include file is in the target(s):
  //brave/browser/android/preferences:java_pref_names_srcjar
  which should somehow be reachable.
___________________
ERROR at //brave/browser/search_engines/search_engine_provider_service_factory.cc:19:11: Include not allowed.
#include "components/search_engines/search_engines_pref_names.h"
          ^----------------------------------------------------
  It is not in any dependency of
  //chrome/browser:browser
  The include file is in the target(s):
  //brave/browser/android/preferences:java_pref_names_srcjar
  which should somehow be reachable.
___________________
ERROR at //brave/browser/profiles/profile_util_unittest.cc:16:11: Can't include this header from here.
#include "components/search_engines/search_engines_pref_names.h"
          ^----------------------------------------------------
  The target:
  //brave/test:brave_unit_tests__library
  is including a file from the target:
  //brave/browser/android/preferences:java_pref_names_srcjar
  
  It's usually best to depend directly on the destination target.
In some cases, the destination target is considered a subcomponent
of an intermediate target. In this case, the intermediate target
should depend publicly on the destination to forward the ability
to include headers.

Dependency chain (there may also be others):
  //brave/test:brave_unit_tests__library -->
  //brave/test:brave_public_test_apk --[private]-->
  //chrome/browser/settings:test_support_java -->
  //chrome/browser/settings:test_support_java__dex --[private]-->
  //chrome/android:chrome_java__header --[private]-->
  //brave/browser/android/preferences:java_pref_names_srcjar