fix: properly initialize FastMCP lifespan in Starlette app

FastMCP requires its lifespan context to be passed to the parent ASGI app
to initialize the StreamableHTTPSessionManager task group. This fix nests
mcp_app.lifespan() inside our custom lifespan to ensure proper initialization
while maintaining our custom startup/shutdown logic.

Resolves: RuntimeError: Task group is not initialized

server.py

@@ -11,6 +11,7 @@ import json
 import logging
 import os
 import sys
+from contextlib import asynccontextmanager
 from pathlib import Path
 from typing import Optional
 
@@ -419,11 +420,15 @@ def create_app() -> Starlette:
     """Create the Starlette application wrapping the MCP server."""
     mcp_app = mcp.http_app()
 
-    async def lifespan(app):
+    # Combine our custom lifespan with MCP's lifespan
-        # Startup
+    @asynccontextmanager
+    async def combined_lifespan(app):
+        # Our startup logic
         logger.info("Plex MCP Server starting up")
-        yield
+        # Nest MCP's lifespan inside ours
-        # Shutdown
+        async with mcp_app.lifespan(app):
+            yield
+        # Our shutdown logic
         logger.info("Plex MCP Server shutting down")
         await plex_client.close()
 
@@ -432,7 +437,7 @@ def create_app() -> Starlette:
             Route("/health", health_check),
             Mount("/", app=mcp_app),
         ],
-        lifespan=lifespan,
+        lifespan=combined_lifespan,
     )