[Meta] New documentation site M1

[tkajtoch]

[!Note] This meta issue tracks the effort of developing the new EUI documentation site only. Check out the Elastic Design System - Milestone 2 for more.

Summary

With the transition to EUI being a part of the Elastic Design System, we want to improve how we create, maintain, and publish our docs.

Goals

• Replace the current website with a shiny new one based on docusaurus (check out the ADR why we chose docusaurus - Elastic internal only) while keeping the content mostly the same
  • We plan to reorganize and rewrite content in further milestones
• Reduce the complexity of writing and maintaining docs by converting them to MDX
• Convert all component examples to use interactive component previews based on react-live (or similar)
• Add full-text search capabilities
• Add Storybook playgrounds for all EUI components

Definition of Done

• The new documentation site is deployed in place of the current one (https://eui.elastic.co) and is publicly accessible
• The new documentation site passes manual QA and accessibility tests
• Buildkite pipelines are updated to build and publish the new website on every EUI release

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### Meta issues
- [ ] https://github.com/elastic/eui/issues/7405
- [ ] https://github.com/elastic/eui/issues/7432
- [ ] https://github.com/elastic/eui/issues/7753
- [ ] https://github.com/elastic/eui/issues/7754
- [ ] https://github.com/elastic/eui/issues/7404
- [ ] https://github.com/elastic/eui/issues/7433

### Issues
- [ ] #7386
- [ ] https://github.com/elastic/eui/issues/7402
- [ ] #7393
- [ ] #7394
- [ ] https://github.com/elastic/eui/issues/7407
- [ ] https://github.com/elastic/eui/issues/7299
- [ ] https://github.com/elastic/eui/issues/7644