[Doc] Unify structured outputs examples (#18196)

Signed-off-by: Aaron Pham <contact@aarnphm.xyz>

docs/features/reasoning_outputs.md

@@ -142,51 +142,6 @@ for chunk in stream:
 
 Remember to check whether the `reasoning_content` exists in the response before accessing it. You could checkout the [example](https://github.com/vllm-project/vllm/blob/main/examples/online_serving/openai_chat_completion_with_reasoning_streaming.py).
 
-## Structured output
-
-The reasoning content is also available in the structured output. The structured output engine like `xgrammar` will use the reasoning content to generate structured output. It is only supported in v0 engine now.
-
-```bash
-vllm serve deepseek-ai/DeepSeek-R1-Distill-Qwen-1.5B --reasoning-parser deepseek_r1
-```
-
-The following is an example client:
-
-```python
-from openai import OpenAI
-from pydantic import BaseModel
-
-# Modify OpenAI's API key and API base to use vLLM's API server.
-openai_api_key = "EMPTY"
-openai_api_base = "http://localhost:8000/v1"
-
-client = OpenAI(
-    api_key=openai_api_key,
-    base_url=openai_api_base,
-)
-
-models = client.models.list()
-model = models.data[0].id
-
-class People(BaseModel):
-    name: str
-    age: int
-
-json_schema = People.model_json_schema()
-
-prompt = ("Generate a JSON with the name and age of one random person.")
-completion = client.chat.completions.create(
-    model=model,
-    messages=[{
-        "role": "user",
-        "content": prompt,
-    }],
-    extra_body={"guided_json": json_schema},
-)
-print("reasoning_content: ", completion.choices[0].message.reasoning_content)
-print("content: ", completion.choices[0].message.content)
-```
-
 ## Tool Calling
 
 The reasoning content is also available when both tool calling and the reasoning parser are enabled. Additionally, tool calling only parses functions from the `content` field, not from the `reasoning_content`.