Uses context URL loader for a search fallback (uplift to 1.24.x)

[brave-builds]

Uplift of #8568
Uplift of #8582
Resolves brave/brave-browser#15399
Resolves brave/brave-browser#15377

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]

test-browser failed on macOS as per https://ci.brave.com/job/pr-brave-browser-pr8568_context_url_loader_search_1.24.x-macos/4/failure-cause-scan-log/. However, these are known intermittent failures and unrelated to this specific PR.

2021-04-22 01:11 Found failed test case: UpgradePath/PreferencesForVersion069WithRewardsAndAdsEnabled_ForSupportedLocale_ForUnsupportedLocale_RewardsShouldBeEnabled_AdsShouldBeEnabled
2021-04-22 01:11 Found failed test case: UpgradePath/PreferencesForVersion12WithRewardsAndAdsEnabled_ForSupportedLocale_ForNewlySupportedLocale_RewardsShouldBeEnabled_AdsShouldBeDisabled

[kjozwiak]

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