refactor(background-agent): remove file persistence, use memory-only

- Remove background_tasks.json persistence (race condition with multiple instances)
- Pure memory-based task management
- Add logging for promptAsync errors
- Remove unused persist/restore methods

🤖 GENERATED WITH ASSISTANCE OF [OhMyOpenCode](https://github.com/code-yeongyu/oh-my-opencode)

src/index.ts

@@ -164,11 +164,7 @@ const OhMyOpenCodePlugin: Plugin = async (ctx) => {
 
   updateTerminalTitle({ sessionId: "main" });
 
-  const backgroundManager = new BackgroundManager(
-    ctx.client,
-    path.join(ctx.directory, ".opencode", "background-tasks.json")
-  );
-  await backgroundManager.restore();
+  const backgroundManager = new BackgroundManager(ctx.client);
 
   const backgroundNotificationHook = createBackgroundNotificationHook(backgroundManager);
   const backgroundTools = createBackgroundTools(backgroundManager, ctx.client);