Fixed wdp infobar is not shown properly after loading search.brave.com

[simonhong]

fix brave/brave-browser#37705

This is partial fix but at least user could see proper wdp info if window is not resized as we need to have this fix in next stable RC.
As we still have wdp infobar layout issue if user changes window width, we ned to fix it as a f/u.

Resolves

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:

See the linked issue

[sangwoo108]

LGTM

[MadhaviSeelam]

Verification PASSED using

Brave | 1.67.30 Chromium: 124.0.6367.60 (Official Build) nightly (64-bit)
-- | --
Revision | e692f54729b32dc0d1a275509f1d77fa07d96363
OS | Windows 11 Version 23H2 (Build 22631.3447)

1. Install 1.67.30
2. launch Brave
3. enter a string (test) in the URL bar or search.brave.com

Confirmed the WDP infobar shown correct text. Help improve Brave Search by sending anonymous usage data...

2024-04-22_10h27_06.mp4