[engineering plugin] GitHub MCP connector stuck: shows "Connected" with unresponsive Disconnect button, but OAuth never completed and tools never load (Cowork, Windows 11)

**Environment**
- Claude Cowork (desktop app), Windows 11
- - Team plan (organization: single-admin)
- - - engineering plugin v1.2.0 from Anthropic & Partners marketplace ("Last updated 6 days ago" as of 2026-07-30)
**Summary**
The GitHub MCP connector bundled with the engineering plugin is in a stuck half-connected state that cannot be cleared from the UI:

1. Settings → Customize → Plugins → Engineering → Connectors shows GitHub as connected, with a **Disconnect** button.
2. 2. The Disconnect button is completely unresponsive (no click feedback, no state change, nothing in UI).
3. 3. In every Cowork session (including brand-new tasks), the server `plugin:engineering:github` never becomes available. Session status reports it as "requires authentication" or "didn't finish connecting." ToolSearch finds no GitHub tools.
4. 4. Verified on the GitHub side (github.com/settings/apps/authorizations and /settings/applications): **no OAuth grant for the GitHub MCP server exists at all** — neither under Authorized GitHub Apps nor Authorized OAuth Apps. The only Anthropic-related grant is the separate "Claude" GitHub App (the claude.ai file-attach integration), which works fine. So the plugin connector's OAuth flow appears to have never completed, while the Cowork UI records it as connected.
**Repro / troubleshooting already attempted (no effect)**
- Full quit of the desktop app (including system tray) and relaunch → Disconnect still unresponsive
- - Toggling the Engineering plugin off and back on from the plugin detail page → connector still shows Connected/Disconnect, still unresponsive
- - - Starting fresh Cowork tasks → server still fails to connect ("didn't finish connecting")
**Expected**
Either the connector completes OAuth and its tools load, or Disconnect works so the stuck record can be cleared and the flow retried.

**Impact**
GitHub tools are unusable in Cowork despite the UI reporting a healthy connection. The misleading "Connected" state cost significant debugging time (user + Claude session) before the half-completed OAuth was identified as the root cause. Notably, all other plugin-bundled connectors in the same install (datadog, pagerduty, linear, notion, figma, etc.) correctly show as not connected / requiring authentication — only GitHub is in this phantom state, which is consistent with an earlier connect attempt that partially wrote state before failing.

*(Filed with the user's permission from a Cowork session via Claude in Chrome, since the connector issue itself prevented in-product reporting paths.)*