[Flyout System] Fix circular dependency of effect and performance issues

[tsullivan]

Summary

Closes #9173

Fix infinite recursion and performance issues in flyout layout mode hook

• Only attach resize listener when child flyout is present
• Extract specific context values instead of depending on entire context object
• Refactor calculation logic using useMemo for better separation of concerns
• Add new tests

This fix can be tested in a Kibana draft PR: elastic/kibana#233806. Run gh pr checkout 233806

Why are we making this change?

Resolve a blocker issue in EUI main.

Screenshots #

Impact to users

QA

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General checklist

• Browser QA
  • Checked in both light and dark modes
  • Checked in both MacOS and Windows high contrast modes
    • (emulate forced colors if you do not have access to a Windows machine.)
  • Checked in mobile
  • Checked in Chrome, Safari, Edge, and Firefox
  • Checked for accessibility including keyboard-only and screenreader modes
• Docs site QA
  • Added documentation
  • Props have proper autodocs (using @default if default values are missing) and playground toggles
  • Checked Code Sandbox works for any docs examples
• Code quality checklist
  • Added or updated jest and cypress tests
  • Updated visual regression tests
• Release checklist
  • A changelog entry exists and is marked appropriately
  • If applicable, added the breaking change issue label (and filled out the breaking change checklist)
  • If the changes unblock an issue in a different repo, smoke tested carefully (see Testing EUI features in Kibana ahead of time)
• Designer checklist
  • If applicable, file an issue to update EUI's Figma library with any corresponding UI changes. (This is an internal repo, if you are external to Elastic, ask a maintainer to submit this request)

[elasticmachine]

💚 Build Succeeded

• Buildkite Build
• Commit: 8557538
• Documentation website
• Storybook

[elasticmachine]

💔 Build Failed

• Buildkite Build
• Commit: 8557538

Failed CI Steps

• 🔨 EUI pull request test

[tsullivan]

Closing in favor of #9178