GitHub plugin should expose organization installation link in connection menu

Summary

The GitHub plugin can report a successful connection even when the ChatGPT Codex Connector GitHub App is installed on zero accounts. The plugin menu does not expose the separate GitHub App installation flow required for private repositories and organizations.

Current behavior

1. Codex Settings → Plugins → GitHub shows the connection as approved and connected.
2. Repository discovery returns only public repositories.
3. Reconnecting the GitHub account reports success but does not prompt for a GitHub App installation target.
4. GitHub's authorization page states: "ChatGPT Codex Connector has not been installed on any accounts you have access to."
5. The organization's Installed GitHub Apps page shows no ChatGPT Codex Connector installation and no pending request.
6. The user must independently discover this direct URL:

https://github.com/apps/chatgpt-codex-connector/installations/new

Requested improvement

Add a visible menu option in the GitHub plugin connection panel:

Install for another GitHub account or organization

The option should open:
https://github.com/apps/chatgpt-codex-connector/installations/new

The connection panel should also distinguish these states:

• GitHub identity authenticated.
• GitHub App installed for a personal account.
• GitHub App installed for each organization.
• No installations found.
• Installation request pending.

If no installation exists, the interface should not present the connector as fully connected without an actionable warning.

Why this matters

GitHub authentication and GitHub App installation are separate steps. Hiding the installation step makes missing private and organization repositories look like a connector failure, and repeated disconnect/reconnect attempts do not resolve it.

Environment

• Codex desktop on macOS.
• ChatGPT Plus.
• GitHub plugin using ChatGPT Codex Connector.
• Personal public repositories were visible.
• Personal private and organization repositories were not visible because no GitHub App installation existed.