[Docs]: Examples cleanup and installation updates

[peterschmidt85]

Summary

• Rework the main docs/examples entry points: make the Examples page tree-based, adjust the docs landing page, and update the related template/CSS pieces that support the new presentation.
• Remove deprecated HuggingFace TGI example content, including the dedicated TGI example page/configs and the navigation/docs references that pointed to them.
• Remove Intel Gaudi example content and Intel-specific DeepSeek example variants that are no longer intended to be part of the visible examples set.
• Update adjacent docs that referenced removed examples, including example indexes plus AMD, vLLM, services, and installation pages, so the docs stay internally consistent after the cleanup.
• Add contributor documentation for docs feature toggles in contributing/DOCS.md, so expensive generators can be disabled while iterating on docs locally.
• Update docs build/generator plumbing in scripts/docs/ to match the revised docs/examples structure.