refactor(exec): route async /task through TaskExecutionService (#95)

[vybe]

Summary

• Delete _execute_task_background (~220 lines of duplicated lifecycle) from src/backend/routers/chat.py. Async /task now delegates to TaskExecutionService.execute_task via a thin _run_async_task_with_persistence wrapper, matching the pattern already used by public.py and internal.py async paths.
• Add three parameters to TaskExecutionService.execute_task(): parent_activity_id, extra_activity_details, slot_already_held. These preserve async-specific behaviors (collaboration parent linkage, parallel_mode flag for the Network-view filter, router-side pre-acquire for the 429-upfront contract).
• Extract a shared _persist_chat_session helper used by both sync and async branches of execute_parallel_task. Guarded on status == SUCCESS to avoid empty assistant messages on FAILED/CANCELLED. Fixes a subscription_id snapshot gap (E3) by passing subscription_id into db.create_task_execution on the async path.

Preserved behavior

• 429-at-capacity returned synchronously — the router pre-acquires the slot and passes slot_already_held=True so the service skips its own acquire but still releases in finally.
• parallel_mode: True on CHAT_START activity — required by src/frontend/src/stores/network.js:255, preserved via extra_activity_details.
• Collaboration activity parenting — CHAT_START now linked to the collaboration activity via the new parent_activity_id param.
• save_to_session + chat_response_ready WebSocket broadcast — moved into the shared helper, still fires on the async path.

Test Plan

• New parity tests in tests/test_parallel_task.py::TestAsyncModeUnifiedExecutor:
  • test_async_mode_at_capacity_returns_429_upfront — confirms 429 contract preserved.
  • test_async_mode_activity_has_parallel_mode_flag — confirms Network-view filter still matches.
• Existing TestAsyncMode* classes still pass (no regression).
• Backend restarts clean; execute_parallel_task async branch serves 202 + execution_id within 1s.
• grep -r "_execute_task_background" returns zero hits in src/.
• Reviewer manual verify: async task from the UI's Tasks tab completes, appears in Network timeline, chat_response_ready WebSocket event fires when save_to_session=True.

Context

First Tier 0 item from docs/planning/ORCHESTRATION_RELIABILITY_2026-04.md. Unblocks the remaining Sprint A fixes (#285, #61, #226, #286, #132, #56) and the Sprint C orchestration work (#260, #294) that layer on the unified executor.

Closes #95.

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