[feat]: Update `<pf-v6-search-input>` for PatternFly v6

Start by reading .claude/ADVICE.md and the existing elements/pf-v5-search-input/ implementation, then inspect the related exports in elements/package.json. Review the docs references, eslint.config.js, and the required demos and tests, including a11ySnapshot assertions. Done means the PFv6 visual design, React SearchInput parity, accessibility checks, reviews, linting, changeset, and staging/pfv6 target are covered.

Summary

Update `` to implement the PatternFly v6 Search input component.

Search field with optional result count, navigation buttons, and expandable/advanced variants.

Prior Art

Requirements

• MUST implement PFv6 visual designs
• MUST provide end-user feature parity with `@patternfly/react-core` SearchInput
• MAY adjust element API to leverage web platform strengths
• SHOULD maintain CSS custom property theming compatibility
• SHOULD expose useful CSS shadow parts

Element-specific considerations

• React `value`, `placeholder`, `onChange` - core text input; integrates with inner `` or native input
• React `resultsCount` - shows "N results" badge next to input; map to `results-count` attribute
• React `onNextClick`, `onPreviousClick` - prev/next result navigation buttons; emit `next` / `previous` events
• React `onClear` - clear button; emit `clear` event
• React `isExpandable` - collapsed to icon button, expands on click; map to `expandable` attribute
• React `expandedBody` / `resetButtonLabel` - expandable advanced search form; verify v6 scope
• React `hint` - hint text shown below/beside input
• ARIA: `type="search"` on inner input; clear button needs `aria-label`; result count should update `aria-live` region

Checklist

Rename

• Delete `elements/pf-v5-search-input/`
• Update `elements/package.json` exports
• Update `eslint.config.js` ignore globs if applicable
• Update docs `_snippets/` and `framework-integration/` references

Implementation

• API per `.claude/ADVICE.md`
• CSS uses v6 tokens, logical properties, nesting
• `@summary` and JSDoc on element class

Demos

• Demos match patternfly.org naming/structure
• CSS custom properties match computed styles from patternfly.org (Chrome MCP)
• Visual parity verified via Chrome MCP

Tests

• Public API covered (attributes, properties, events, slots)
• `a11ySnapshot` assertions

Reviews

• `/review-api` passes
• `/review-demos` passes
• `/review-a11y` passes
• `eslint` and `stylelint` pass

Ship

• Add changeset
• PR targets `staging/pfv6` (NOT `main`)