Local MCP servers fail with 'Not ready after 60 seconds' even though the remote connection succeeds (shared-pool readiness bug)

# Title
Local MCP servers fail with "Not ready after 60 seconds" / "Request timed out" even though the remote connection succeeds (shared-pool readiness bug)

# Environment
- App: Claude Desktop (Cowork build), Windows 11 Enterprise
- Update channel: `autoUpdatesChannel: "latest"` (auto-updates enabled)
- Config: 5 local MCP servers defined in `claude_desktop_config.json` (3 remote via `mcp-remote` + 2 local stdio: playwright, context7)
- Network: corporate proxy (reachable, verified via `Test-NetConnection`), internal domain reachable directly in browser

# Summary
All 5 configured local MCP servers (confluence-mcp, jira-mcp, github-mcp, playwright, context7) suddenly started failing this morning with no config changes since the previous day, when all 5 were working. The same MCP configs work correctly via the `claude` CLI (Claude Code) on the same machine at the same time, ruling out network/proxy/token/cert issues.

# Steps to reproduce
1. Configure any MCP server via `mcpServers` in `claude_desktop_config.json` (works fine one day)
2. Restart the app (auto-update likely applies here)
3. Open Settings → 계정 → 로컬 MCP 서버 — all servers show ⚠️ 실패 with "Couldn't start for Cowork and Code sessions. Error: Request timed out"
4. In parallel, run `claude mcp list` in a terminal with the identical server definitions — all connect successfully (✔ Connected)

# Expected behavior
Local MCP servers configured in `claude_desktop_config.json` connect successfully, consistent with the `claude` CLI behavior using the same config.

# Actual behavior
Per `%LOCALAPPDATA%\Claude\Logs\mcp-server-confluence-mcp.log`, the underlying `mcp-remote` process actually succeeds in connecting to the remote server:

```
[27284] Connected to remote server using StreamableHTTPClientTransport
[27284] Local STDIO server running
[27284] Proxy established successfully between local STDIO and remote StreamableHTTPClientTransport
```

But shortly after, the app's internal session-pool layer reports a timeout and shuts the server down before the UI/session ever sees it as ready:

```
[info] Server transport closed (renderer released port); shutting down server
[error] Couldn't start for Cowork and Code sessions. Error: Not ready after 60 seconds; the sessions waiting for it started without it { metadata: { context: 'shared-pool', ... } }
[error] Couldn't start for Cowork and Code sessions. Error: Request timed out { metadata: { context: 'shared-pool', ... } }
```

This indicates the underlying connection succeeds, but the app's shared MCP-process-pool coordination (used across Cowork/Code sessions in the same app) fails to propagate the "ready" signal within its 60s window, and reports a false failure to the UI.

# Notes
- This reproduces identically for all 5 configured servers (3 remote HTTP via mcp-remote, 2 local stdio), suggesting the bug is in the shared-pool layer itself rather than any individual server's transport/auth.
- Fully quitting (including tray icon) and relaunching the app did not resolve it.
- Suspect this coincides with an overnight auto-update (`autoUpdatesChannel: "latest"`), since nothing in the local config changed between the last known-good state and this failure.

# Attachments
- (attach a redacted excerpt of `mcp-server-confluence-mcp.log` around the failure timestamp — double-check no `x-auth`/`Authorization` header values are present before attaching)

Reproduce with the five entries in claude_desktop_config.json and compare the desktop logs in %LOCALAPPDATA%\Claude\Logs\mcp-server-confluence-mcp.log with `claude mcp list`. Trace the shared-pool readiness path around the reported 60-second timeout and verify that successfully connected local and remote servers reach the ready state before shutdown.

Written by the indexing model from the issue text.