Docs: rewrite the public home and getting-started flow

[nerdchanii]

Summary

• rewrite the public docs home page as a user-facing landing page
• add install guidance and a minimal provider-based React usage path near the top
• point readers to the guide, examples, API, Next.js, compatibility, and migration pages in a natural order

Type

• API
• Runtime
• Testing
• Docs
• Tooling
• Packaging/release

Priority

• P0 - Critical
• P1 - High
• P2 - Medium
• P3 - Low
• P4 - Backlog

Verification

• Tests added or updated
• Typecheck passes
• Build passes
• Docs/examples updated if needed

Release Intent

• Changeset added
• No release needed (docs/project-only or tooling-only and does not affect consumers)

Notes

Related issue: Closes #117

Verification:

• npm run docs:build
• npm run lint

[nerdchanii]

@codex review

Context:

• Issue: Docs: Rewrite public home and getting started flow #117, milestone: post-v1 stabilization

[chatgpt-codex-connector]

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[chatgpt-codex-connector]

Disable sending until the WebSocket is open

When this getting-started snippet is copied against a real endpoint, ChatMessages renders as soon as the SocketStore is constructed, so the button can be clicked while the WebSocket is still CONNECTING or after the placeholder endpoint fails. useSocket forwards the click directly to store.send, and the core socket-store send contract throws ERR_SOCKET_NOT_OPEN unless readyState === 1, so the first click can crash the example; gate the button/rendering on an open/onConnect state or otherwise handle the not-open case.

Useful? React with 👍 / 👎.