feat(website): update for deeply nested examples

[pront]

Summary

Ported #19598 to this one. So we can get a site preview.

Vector configuration

How did you test this PR?

Change Type

• Bug fix
• New feature
• Non-functional (chore, refactoring, docs)
• Performance

Is this a breaking change?

• Yes
• No

Does this PR include user facing changes?

• Yes. Please add a changelog fragment based on our guidelines.
• No. A maintainer will apply the no-changelog label to this PR.

References

Notes

• Please read our Vector contributor resources.
• Do not hesitate to use @vectordotdev/vector to reach out to us regarding this PR.
• Some CI checks run only after we manually approve them.
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  • Alternatively, we recommend running the following locally before pushing to the remote branch:
    • cargo fmt --all
    • cargo clippy --workspace --all-targets -- -D warnings
    • cargo nextest run --workspace (alternatively, you can run cargo test --all)
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  • Feel free to push as many commits as you want. They will be squashed into one before merging.
  • For example, you can run git merge origin master and git push.
• If this PR introduces changes Vector dependencies (modifies Cargo.lock), please
  run cargo vdev build licenses to regenerate the license inventory and commit the changes (if any). More details here.

[github-actions]

Your preview site for the vector.dev will be ready in a few minutes, please allow time for it to build.

Heres your preview link:
vector.dev preview

[github-actions]

Your preview site for the VRL Playground will be ready in a few minutes, please allow time for it to build.

Heres your preview link:
VRL Playground preview

[datadog-vectordotdev]

Datadog Report

Branch report: devin-ford-website-4216-2
Commit report: d185812
Test service: vector

✅ 0 Failed, 7 Passed, 0 Skipped, 18.92s Total Time