[BUG] GitHub connector shows "Connected" and app has correct repo access, but no GitHub tools are exposed to Claude in conversation (search_mcp_registry returns empty)

### Preflight Checklist

- [x] I have searched [existing issues](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20state%3Aopen%20label%3Abug) and this hasn't been reported yet
- [x] This is a single bug report (please file separate reports for different bugs)
- [x] I am using the latest version of Claude Code

### What's Wrong?

The GitHub Integration connector shows "Connected" in claude.ai → Settings → Connectors. The Claude GitHub App is confirmed installed at github.com/settings/installations, with repository access correctly granted (verified via Configure — all four of my repos are selectable). The "Add content from GitHub" picker in a Project chat correctly lists and lets me select repositories.

However, no GitHub tool is ever exposed to Claude in the actual conversation. Searching the tool registry for "github" or "repository" returns no GitHub-specific results — only unrelated tools (Vercel, Lovable) that happen to reference repositories for deployment purposes.

### What Should Happen?

With the GitHub connector showing "Connected" and the Claude GitHub App installed with valid repository access, Claude should have a functioning GitHub tool available in the conversation (e.g. list files, read file/repository content). This should work the same way other connected apps like Google Drive or Gmail expose their tools once connected.

### Error Messages/Logs

```shell
No error message or exception is thrown. The absence is silent — the GitHub tool simply does not appear in Claude's available tool list, with no accompanying log output, console error, or failure notification anywhere in the UI.
```

### Steps to Reproduce

1. In claude.ai Settings → Connectors, confirm GitHub Integration shows "Connected"
2. At github.com/settings/installations, confirm the Claude GitHub App is installed and click Configure to verify repository access (in my case: "All repositories", all four repos selectable)
3. In a Project chat on claude.ai, open "Add content from GitHub" and confirm the repo picker populates correctly and a repository can be selected
4. In the same conversation, ask Claude to check what GitHub-related tools are available, or ask it to read content from the linked repository
5. No GitHub tool is available to Claude — searching the tool registry for "github" or "repository" returns no GitHub-specific results, only unrelated tools (e.g. Vercel, Lovable) that reference repositories incidentally for deployment

### Claude Model

Sonnet (default)

### Is this a regression?

I don't know

### Last Working Version

_No response_

### Claude Code Version

N/A — not using Claude Code CLI, this is claude.ai web (Projects)

### Platform

Anthropic API

### Operating System

Windows

### Terminal/Shell

PowerShell

### Additional Information

Note: This report concerns claude.ai web chat (Projects), not Claude Code CLI. The Claude Model, Claude Code Version, and Terminal/Shell fields above don't cleanly apply — filled with placeholder values because the form requires them. Browser: Falkon,Chrome,Claude desktop. It doesn't work on my home windows pc or on my work Linux.

No repository file, test, or code entry point is identified in the report; start by reproducing the connector and tool-registry steps in claude.ai Projects with the listed browsers and repository permissions. Done means a connected GitHub account exposes GitHub-specific tools in the conversation, or the UI reports the reason they are unavailable.

Written by the indexing model from the issue text.