Backup search API (Uplift to 1.24.x)

[bbondy]

Fixes:

• make an API for backup search results in the chrome object brave-browser#14910
• Move backup results logic to service workers brave-browser#15217
• Re-enable browser tests on brave search functionality brave-browser#15295
• Search API changes brave-browser#15323
• Rename self param to safe for search brave-browser#15333
• Search country code should use gl= and not c= brave-browser#15130
• Networking code for opt-in backup search results brave-browser#15059

Uplift of:

• Brave search param changes and extra tests #8537
• Moves backup results function to service workers #8493
• Injects window.chrome.fetchBackupResults for opt-in backup results #8338
• Rename self param to safe for search #8551
• Fix param for country code #8439
• Backup search networking code #8407

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]

Going to restart Linux as test-browser failed and I've never seen GeminiAPIBrowserTest.GetAccessTokenServerError before. It's most likely not related to any of the above changes but we'll see if this is an intermittent failure or a new one that's occurring often.

post-init failed due to a audit-dep warning as per https://ci.brave.com/job/pr-brave-browser-search-sw-1.24.x-post-init/1/execution/node/145/log/. However, this is a known issue and not related to this specific PR.

[kjozwiak]

Uplift into 1.24.x approved after deliberating with @brave/uplift-approvers.