Add Brave Search as an alternative search (uplift to 1.26.x)

[brave-builds]

Uplift of #8966
Resolves brave/brave-browser#15663

Pre-approval checklist:

• You have tested your change on Nightly.
• This contains text which needs to be translated.
  • There are more than 7 days before the release.
  • I've notified folks in #l10n on Slack that translations are needed.
• The PR milestones match the branch they are landing to.

Pre-merge checklist:

• You have checked CI and the builds, lint, and tests all pass or are not related to your PR.

Post-merge checklist:

• The associated issue milestone is set to the smallest version that the changes is landed on.

[kjozwiak]

Restarting iOS as build` failed as per https://ci.brave.com/job/pr-brave-browser-pr8966_bsc-brave-search_1.26.x-ios/1/execution/node/139/log/.

FAILED: gen/brave/vendor/brave-ios/ads.mojom.objc.h gen/brave/vendor/brave-ios/ads.mojom.objc+private.h gen/brave/vendor/brave-ios/ads.mojom.objc.mm gen/brave/vendor/brave-ios/objc_templates_bytecode/ads_objc_templates.zip 

Restarting Linux as test-browser failed as per https://ci.brave.com/job/pr-brave-browser-pr8966_bsc-brave-search_1.26.x-linux/1/testReport/. I haven't seen these before and @bsclifton mentioned everything passed on macOS & Win locally before uplifting. Going to restart just to make sure these are intermittent and the above PR hasn't caused these tests to fail.

linux / test-browser / BraveContentBrowserClientTest.CanLoadChromeURL
linux / test-browser / BraveToolbarViewTest.AvatarButtonIsShownMultipleProfiles
linux / test-browser / BraveWelcomeUIBrowserTest.StartupURLTest
linux / test-browser / ScriptTriggerWidevinePermissionRequestBrowserTest.SuggestPermissionIfWidevineDetected

[kjozwiak]

Uplift into 1.26.x approved after deliberating with @brave/uplift-approvers. QA has verified the PR on Nightly as per #8966 (comment).