[Bug] claude mcp list reports stale 'Connected' after MCP server crash

Bug Report

Note: Filed via gh because /feedback is broken — see #41363.

Description

claude mcp list / /mcp continues to report ✓ Connected for an MCP server that has crashed and whose tools have been deregistered from the session.

Steps to Reproduce

1. Configure browser-mcp via npx @browsermcp/mcp
2. Start Claude Code — /mcp shows ✓ connected
3. Wait for the server to crash (browser-mcp crashes reliably within minutes)
4. Observe: all mcp__browser-mcp__* tools permanently deregistered — system reminder confirms "The following deferred tools are no longer available (their MCP server disconnected)"
5. Run /mcp — still shows ✓ Connected

Expected Behavior

Status should reflect actual server state. If tools have been deregistered, status should show failed/disconnected immediately.

Actual Behavior

Stale health check. Tools gone, status green. Eventually updates to ✗ failed after significant lag.

Impact

Users cannot tell whether an MCP server is healthy. Debugging requires --debug logs because the status display lies.

Related Issues

• #41363 — /feedback broken (why this is filed via gh)
• #41099, #41101, #41104 — Computer Use MCP permission and browser automation issues

Environment

• macOS Darwin 25.2.0
• Claude Code 2.1.88, Opus 4.6