Feature request: machine-readable + live MCP connection status for wrapper/IDE integrations

Problem

I build a GUI that runs claude inside a PTY for users. A recurring pain point is that MCP servers silently disconnect mid-session and users don't notice until a tool call fails. Today there is no supported way for a wrapper to know the live MCP connection state of a running session:

• claude mcp list health-checks, but only prints human-readable glyphs (✓ / ✗ / ! / ⏸) with no --json, so we must regex terminal text.
• It also runs as a separate probe process, so it reflects config-level reachability, not the actual sockets of the running interactive session.
• The /mcp TUI view has the real live status, but it's only rendered in the TUI and can't be exported without screen-scraping.
• SDK-side health monitoring was requested in #6846 and closed as "not planned".

Why the existing options aren't enough

• Out-of-band probing is heavy (spawns real connections each time) and is affected by health-checker correctness bugs (#7404 false "Failed to connect", #41367 stale "Connected" after a crash).
• Screen-scraping the TUI is brittle and hijacks the user's terminal.

Proposed (in order of cost-to-implement)

1. claude mcp list --json / claude mcp get <name> --json — emit the status you already compute as structured data, e.g.:

{
  "servers": [
    { "name": "context7", "scope": "user", "type": "stdio",
      "status": "connected", "tool_count": 12, "error": null },
    { "name": "remote-api", "scope": "project", "type": "http",
      "status": "failed", "tool_count": 0,
      "error": "HTTP 503", "http_status": 503 }
  ]
}

(status enum: connected | failed | needs_auth | pending_approval | connecting)
This overlaps with #6574 — happy to consolidate.

2. Live status for a running session (the real ask): surface per-server connect / disconnect / reconnect as events in headless / --output-format stream-json mode, e.g. a {"type":"mcp_status", ...} line, and/or write a per-session status file under the session dir that is updated on state change, so a wrapper can poll/watch it without scraping the TUI.

3. Make the health check trustworthy — the data above is only useful if the connectivity check is correct; see #7404 and #41367.

Use case

Third-party IDE/GUI integrations that embed claude and want to show a "MCP 3/4 connected" indicator and warn the user the moment a server drops.

Related: #6846 (closed), #6574, #7404, #41367, #29035.