fix: enforce single playing game and clean up stale shard monitors

Mark games as 'played' when shard detects game.ended or room_closed.
Stop old shard monitors before demoting previous playing games on new
game add or status change. Sync frontend playingGame state with the
games list on every refresh to prevent stale UI. Use terminate() for
probe connections to prevent shard connection leaks.

Made-with: Cursor

frontend/src/pages/Picker.jsx

@@ -1,4 +1,4 @@
-import React, { useState, useEffect, useCallback } from 'react';
+import React, { useState, useEffect, useCallback, useRef } from 'react';
 import { useNavigate } from 'react-router-dom';
 import { useAuth } from '../context/AuthContext';
 import api from '../api/axios';
@@ -952,17 +952,33 @@ function SessionInfo({ sessionId, onGamesUpdate, playingGame, setPlayingGame })
   const [editingPlayerCount, setEditingPlayerCount] = useState(null);
   const [newPlayerCount, setNewPlayerCount] = useState('');
 
+  const playingGameRef = useRef(playingGame);
+  playingGameRef.current = playingGame;
+
   const loadGames = useCallback(async () => {
     try {
       const response = await api.get(`/sessions/${sessionId}/games`);
-      // Reverse chronological order (most recent first)
-      setGames(response.data.reverse());
+      const freshGames = response.data;
+      setGames(freshGames.reverse());
+
+      const currentPlaying = freshGames.find(g => g.status === 'playing');
+      const prev = playingGameRef.current;
+      if (currentPlaying) {
+        if (!prev || prev.id !== currentPlaying.id || prev.player_count !== currentPlaying.player_count) {
+          setPlayingGame(currentPlaying);
+        }
+      } else if (prev) {
+        const still = freshGames.find(g => g.id === prev.id);
+        if (!still || still.status !== 'playing') {
+          setPlayingGame(null);
+        }
+      }
     } catch (err) {
       console.error('Failed to load session games');
     } finally {
       setLoading(false);
     }
-  }, [sessionId]);
+  }, [sessionId, setPlayingGame]);
 
   useEffect(() => {
     loadGames();