Fixes extensions install prompt (1.10.x).

[mkarolin]

Fixes brave/brave-browser#10113
Uplift from #5772

Submitter Checklist:

• Submitted a ticket for my issue if one did not already exist.
• Used Github auto-closing keywords in the commit message.
• Added/updated tests for this change (for new code or code which already has tests).
• Verified that these changes build without errors on
  • Android
  • iOS
  • Linux
  • macOS
  • Windows
• Verified that these changes pass automated tests (unit, browser, security tests) on
  • iOS
  • Linux
  • macOS
  • Windows
• Verified that all lint errors/warnings are resolved (npm run lint)
• Ran git rebase master (if needed).
• Ran git rebase -i to squash commits (if needed).
• Tagged reviewers and labelled the pull request as needed.
• Request a security/privacy review as needed.
• Add appropriate QA labels (QA/Yes or QA/No) to include the closed issue in milestone
• Public documentation has been updated as necessary. 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-issues-with-tracking-protection
  • https://github.com/brave/brave-browser/wiki/QA-Guide

Test Plan:

Reviewer Checklist:

• New files have MPL-2.0 license header.
• Request a security/privacy review as needed.
• Adequate test coverage exists to prevent regressions
• Verify test plan is specified in PR before merging to source

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.

[LaurenWags]

Linux failed due to known issue - https://ci.brave.com/job/pr-brave-browser-maxk-fix-extensions-prompt-1.10.x/1/execution/node/407/log/

15:57:43  [  FAILED  ] BraveRewardsBrowserTest.CheckIfReconcileWasResetEmpty, where TypeParam =  and GetParam() =  (30882 ms)
15:57:43  [571/571] BraveRewardsBrowserTest.CheckIfReconcileWasResetEmpty (30935 ms)
15:57:43  1 test failed:
15:57:43      BraveRewardsBrowserTest.CheckIfReconcileWasResetEmpty (../../brave/components/brave_rewards/browser/rewards_service_browsertest.cc:2941)

macOS failed due to known issue - https://ci.brave.com/job/pr-brave-browser-maxk-fix-extensions-prompt-1.10.x/1/execution/node/530/log/

16:28:06  [  FAILED  ] BraveRewardsBrowserTest.CheckIfReconcileWasResetEmpty, where TypeParam =  and GetParam() =  (31528 ms)
16:28:06  [566/566] BraveRewardsBrowserTest.CheckIfReconcileWasResetEmpty (31687 ms)
16:28:06  1 test failed:
16:28:06      BraveRewardsBrowserTest.CheckIfReconcileWasResetEmpty (../../brave/components/brave_rewards/browser/rewards_service_browsertest.cc:2941)

[LaurenWags]

Uplift into 1.10.x approved after deliberating with @brave/uplift-approvers. As per #5787 (comment), the failures on CI are known issues and not blocking uplift.