refactor(features/background-agent): remove dead code submodules

Delete 15 unused files in background-agent module:

- background-task-completer.ts

- format-duration.ts

- message-dir.ts

- parent-session-context-resolver.ts

- parent-session-notifier.ts (and its test file)

- result-handler-context.ts

- result-handler.ts

- session-output-validator.ts

- session-task-cleanup.ts

- session-todo-checker.ts

- spawner/background-session-creator.ts

- spawner/concurrency-key-from-launch-input.ts

- spawner/spawner-context.ts

- spawner/tmux-callback-invoker.ts

Update index.ts barrel and manager.ts/spawner.ts imports

Ultraworked with [Sisyphus](https://github.com/code-yeongyu/oh-my-opencode)

Co-authored-by: Sisyphus <clio-agent@sisyphuslabs.ai>

src/features/background-agent/manager.ts

@@ -268,7 +268,7 @@ export class BackgroundManager {
       body: {
         parentID: input.parentSessionID,
         title: `${input.description} (@${input.agent} subagent)`,
-      } as any,
+      } as Record<string, unknown>,
       query: {
         directory: parentDirectory,
       },