### What do you see as an issue?

The [/dags/{dag_id}/details](https://airflow.apache.org/docs/apache-airflow/stable/stable-rest-api-ref.html#operation/post_set_task_instances_state) endpoint of the REST API does not return all of the keys that are listed in the documentation. If I run `curl -X GET localhost:8080/api/v1/dags/{my_dag}/details`, then compare the results with the results in the documentation, you can see the following missing keys:

```python
>>> for key in docs.keys():
...     if not key in actual.keys():
...             print(key)
...
root_dag_id
last_parsed_time
last_pickled
last_expired
scheduler_lock
timetable_description
has_task_concurrency_limits
has_import_errors
next_dagrun
next_dagrun_data_interval_start
next_dagrun_data_interval_end
next_dagrun_create_after
template_search_path
```

### Solving the problem

Either remove these keys from the documentation or fix the API endpoint

### Anything else

_No response_

### Are you willing to submit PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

external_tasks = session.scalars(query.where(TI.operator == ExternalTaskMarker.__name__))
            for ti in external_tasks:
                ti_key = ti.key.primary
                if ti_key in visited_external_tis:
                    continue
                visited_external_tis.add(ti_key)
                task: ExternalTaskMarker = cast(ExternalTaskMarker, copy.copy(self.get_task(ti.task_id)))
                ti.task = task
                if max_recursion_depth is None:
                    
                    
                    max_recursion_depth = task.recursion_depth
                if recursion_depth + 1 > max_recursion_depth:
                    
                    raise AirflowException(
                        f"Maximum recursion depth {max_recursion_depth} reached for "
                        f"{ExternalTaskMarker.__name__} {ti.task_id}. "
                        f"Attempted to clear too many tasks or there may be a cyclic dependency."
                    )
                ti.render_templates()
                external_tis = session.scalars(
                    select(TI)
                    .join(TI.dag_run)
                    .where(
                        TI.dag_id == task.external_dag_id,
                        TI.task_id == task.external_task_id,
                        DagRun.execution_date == pendulum.parse(task.execution_date),
                    )
                )
                for tii in external_tis: