### Apache Airflow version

main (development)

### What happened

https://github.com/apache/airflow/pull/27755 added the triggerer to the rest api health endpoint, but not the main one served on `/health`.

### What you think should happen instead

As documented [here](https://airflow.apache.org/docs/apache-airflow/2.6.1/administration-and-deployment/logging-monitoring/check-health.html#webserver-health-check-endpoint), the `/health` endpoint should include triggerer info like shown on `/api/v1/health`.

### How to reproduce

Compare `/api/v1/health` and `/health` responses.

### Operating System

mac os

### Versions of Apache Airflow Providers

_No response_

### Deployment

Virtualenv installation

### Deployment details

_No response_

### 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)

TaskInstance.run_id,
            TaskInstance.state,
            TaskInstance._try_number,
            func.min(TaskInstanceNote.content).label("note"),
            func.count(func.coalesce(TaskInstance.state, sqla.literal("no_status"))).label("state_count"),
            func.min(TaskInstance.start_date).label("start_date"),
            func.max(TaskInstance.end_date).label("end_date"),
        )
        .join(TaskInstance.task_instance_note, isouter=True)
        .filter(
            TaskInstance.dag_id == dag.dag_id,
            TaskInstance.run_id.in_([dag_run.run_id for dag_run in dag_runs]),
        )
        .group_by(TaskInstance.task_id, TaskInstance.run_id, TaskInstance.state, TaskInstance._try_number)
        .order_by(TaskInstance.task_id, TaskInstance.run_id)
    )
    grouped_tis = {task_id: list(tis) for task_id, tis in itertools.groupby(query, key=lambda ti: ti.task_id)}
    def task_group_to_grid(item, grouped_tis, *, is_parent_mapped: bool):
        if not isinstance(item, TaskGroup):
            def _get_summary(task_instance):
                return {
                    "task_id": task_instance.task_id,
                    "run_id": task_instance.run_id,
                    "state": task_instance.state,
                    "start_date": task_instance.start_date,
                    "end_date": task_instance.end_date,
                    "try_number": wwwutils.get_try_count(task_instance._try_number, task_instance.state),
                    "note": task_instance.note,
                }
            def _mapped_summary(ti_summaries):