Story 2406: Update Search Bar with No-JS design

[julhoang]

Issue: #2406

Summary & Context

This ticket doesn't require any BE changes as the search component is already connected to Algolia and correctly run queries across two indexes: Libraries and Learn. The actual content crawling, indexing and search logic is handled by this Boost Gecko repo: https://github.com/cppalliance/boost-gecko. This PR handles small UI updates to the No-JavaScript state.

• Figma link: We currently don't have a dedicated disabled state design for the search bar, but we're following this style:

Screenshot	Link
	See Figma reference

• Link to components/page: http://localhost:8000 (on the navbar of any pages!)

Changes

• Updated the component to add a design state for when JS is disabled.
• Extracted the search bar into its own component for code organization and improved documentation.

‼️ Risks & Considerations ‼️

None

Screenshots

Regular Case (with JS)	No JavaScript	No JavaSript on mobile

Self-review Checklist

• Tag at least one team member from each team to review this PR
• Link this PR to the related GitHub Project ticket

Frontend

• UI implementation matches Figma design
• Tested in light and dark mode
• Responsive / mobile verified
• Accessibility checked (keyboard navigation, etc.)
• Ensure design tokens are used for colors, spacing, typography, etc. – No hardcoded values
• Test without JavaScript (if applicable)
• No console errors or warnings

[herzog0]

Behavior is fine. I just think that from a UI/UX perspective, if the field is disabled we should make that more clear with the standard disabled colors we have in the Figma foundational specs, and remove the cursor pointer.
I'm approving though, as there're no designs for that!

[ycanales]

Works without issues. I agree with @herzog0 that design could be improved maybe removing the hover state. Approving.

[julhoang]

I totally agree with you both @herzog0 and @ycanales that the design can be improved 😄! We're leaving it for now to an another ticket as we wait for design support (as confirmed here).