Refactor Breadcrumb to be more SSR friendly

[adierkens]

Refactors the Breadcrumbs component to extract responsive overflow logic into a standalone hook (useBreadcrumbsResponsive) and separate overflow menu component, improving code organization and enabling SSR-friendly usage patterns.

Changelog

Changed

Refactor: Extract responsive breadcrumbs logic

• New useBreadcrumbsResponsive hook - Encapsulates all overflow calculation logic (resize observation, width measurements, visibility state) into a reusable hook that can be used independently for custom
  implementations
• New BreadcrumbsOverflowMenu component - Extracts the overflow menu dropdown into a standalone component
• Adds new responsive prop to Breadcrumbs - allows opting out of automatic overflow handling for manual control
• Exports useBreadcrumbsResponsive from the package for consumers who need SSR-friendly conditional rendering

Fix: Prevent overflow during SSR

• Adds overflow: hidden to breadcrumbs container with menu or menu-with-root overflow modes to prevent content from overflowing before JavaScript calculates the overflow menu

Rollout strategy

• Patch release
• Minor release
• Major release; if selected, include a written rollout or migration plan
• None; if selected, include a brief description as to why

Testing & Reviewing

Merge checklist

• Added/updated tests
• Added/updated documentation
• Added/updated previews (Storybook)
• Changes are SSR compatible
• Tested in Chrome
• Tested in Firefox
• Tested in Safari
• Tested in Edge
• (GitHub staff only) Integration tests pass at github/github (Learn more about how to run integration tests)

[changeset-bot]

🦋 Changeset detected

Latest commit: 12c1d30

The changes in this PR will be included in the next version bump.

This PR includes changesets to release 1 package

Name	Type
@primer/react	Minor

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[github-actions]

👋 Hi, this pull request contains changes to the source code that github/github-ui depends on. If you are GitHub staff, test these changes with github/github-ui using the integration workflow. Or, apply the integration-tests: skipped manually label to skip these checks.