Rebuild the docs around tested examples; shrink README.v2.md to a pitch (#2978)

docs_src/asgi/tutorial002.py

@@ -0,0 +1,27 @@
+from collections.abc import AsyncIterator
+from contextlib import asynccontextmanager
+
+from starlette.applications import Starlette
+from starlette.routing import Mount
+
+from mcp.server import MCPServer
+
+mcp = MCPServer("Notes")
+
+
+@mcp.tool()
+def add_note(text: str) -> str:
+    """Save a note."""
+    return f"Saved: {text}"
+
+
+@asynccontextmanager
+async def lifespan(app: Starlette) -> AsyncIterator[None]:
+    async with mcp.session_manager.run():
+        yield
+
+
+app = Starlette(
+    routes=[Mount("/", app=mcp.streamable_http_app())],
+    lifespan=lifespan,
+)