[Search results] Use GlSearchboxByClick to have correct styles, and improve usability

Background

This issue represents a finding from our work to audit features belonging to groupglobal search. See the analysis in Figma for more information. See the overview video for guidance on how to review this analysis.

Problem to solve

• Use GlSearchboxByClick to have correct styles, and improve usability
• This will prevent the Search results view from adapting properly to dark mode.

Proposal

• Address the finding on the Search results view. The specific solution may require discussion between design and engineering.
• Designers from groupdesign system have the following guidance:
  • This impact all Advanced Search pages
• View finding black#5 in Figma

Screenshots

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