[Toolbar]: Fix close search button

[fallaciousreasoning]

Resolves brave/brave-browser#42678

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 issue

[mkarolin]

LGTM

[brave-builds]

A Storybook has been deployed to preview UI for the latest push

[brave-builds]

Released in v1.75.76

[LaurenWags]

Verified with

Brave	1.75.76 Chromium: 132.0.6834.33 (Official Build) nightly (arm64) 
Revision	d1eb5f0ad0eb32f6bc8a8b72e2b6a7b34127706a
OS	macOS Version 15.2 (Build 24C5089c)

Reproduced the issue using STR from brave/brave-browser#42678 and 1.75.72 Chromium: 132.0.6834.15:

72.mov

Using 1.75.76 Chromium: 132.0.6834.33 and same STR confirmed that clicking "x" icon in the settings search bar closes the search bar as expected. Confirmed clicking "x" closes the search bar whether it is empty (no text typed) or if the search bar has text entered in it:

76.mov