Migrate the chat completion app endpoint to the OpenAI Responses API

## StatGPT Backend version

`latest`

## What is the problem this feature will solve?

The app backend currently exposes its conversational interface through the Chat Completions API surface (`add_chat_completion_with_dependencies` / `ChatCompletion` in `statgpt/app/application/app_factory.py`). The Chat Completions API is effectively in maintenance mode upstream, while the **OpenAI Responses API** is the forward-looking interface and offers a richer, stateful model for agentic flows (built-in tool calls, reasoning items, structured streaming events, and server-side state).

Staying on Chat Completions limits our ability to adopt these capabilities and keeps us on an API surface that no longer receives new features.

## What is the proposed feature or solution?

Migrate the app endpoint to serve the OpenAI Responses API instead of (or alongside) Chat Completions.

> [!IMPORTANT]
> **Prerequisite / blocker:** `aidial-sdk` does **not** currently support the Responses API. SDK support must be added upstream first before this migration can proceed. This issue depends on that work.

High-level scope once the SDK supports it:

- Replace/augment the chat-completion registration in `DialAppFactory` with a Responses-based endpoint.
- Adapt the channel completion implementations (`ChannelCompletion`, `ChannelOnboardingCompletion`) to the Responses request/response and streaming model.
- Map our existing tool-calling agent output (tool calls, intermediate results, stages) onto Responses event/item types.
- Preserve existing DIAL integration behavior (configuration endpoint, heartbeat, telemetry, middleware).

## What alternatives have you considered?

- **Stay on Chat Completions** — no upstream SDK work needed, but forgoes Responses API capabilities and remains on a maintenance-mode surface.
- **Dual-serve both APIs during a transition** — lower migration risk, at the cost of maintaining two code paths.