fix(chess): send game updates to move sender + responsive mobile redesign

Bun's ws.publish() excludes the sender, so the player making a move never
received the GAME_UPDATE with the new FEN — causing pieces to snap back.
Added ctx.send() alongside ctx.publish() for GAME_UPDATE and GAME_ENDED.

Also redesigned the panel for mobile: hamburger drawer sidebar, responsive
chess board sizing via ResizeObserver, stacked layouts on small screens,
and touch-friendly modals/controls across lobby and game pages.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

api/src/games/ws-handler.ts

@@ -77,15 +77,19 @@ export function handleGameMessage(msg: GameWsClientMessage, ctx: WsContext): voi
             }
 
             const spectatorView = roomManager.getSpectatorView(msg.roomId);
-            ctx.publish(`room:${msg.roomId}`, JSON.stringify({ type: "GAME_UPDATE", roomId: msg.roomId, state: spectatorView }));
+            const updateMsg = JSON.stringify({ type: "GAME_UPDATE", roomId: msg.roomId, state: spectatorView });
+            ctx.publish(`room:${msg.roomId}`, updateMsg);
+            ctx.send(updateMsg);
 
             if (result.gameOver) {
-                ctx.publish(`room:${msg.roomId}`, JSON.stringify({
+                const endedMsg = JSON.stringify({
                     type: "GAME_ENDED",
                     roomId: msg.roomId,
                     winner: result.gameOver.winner,
                     reason: result.gameOver.reason,
-                }));
+                });
+                ctx.publish(`room:${msg.roomId}`, endedMsg);
+                ctx.send(endedMsg);
                 ctx.publish("lobby", JSON.stringify({ type: "ROOM_LIST_UPDATE", rooms: roomManager.listRooms() }));
             }
             break;