### 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)

"""
    Parse datetime and return error message for invalid dates.
    :param v: the string value to be parsed
    :param allow_empty: Set True to return none if empty str or None
    """
    if allow_empty is True and not v:
        return None
    try:
        return timezone.parse(v)
    except (TypeError, ParserError):
        abort(400, f"Invalid datetime: {v!r}")
def node_dict(node_id, label, node_class):
    return {
        "id": node_id,
        "value": {"label": label, "rx": 5, "ry": 5, "class": node_class},
    }
def dag_to_grid(dag: DagModel, dag_runs: Sequence[DagRun], session: Session):
    """
    Create a nested dict representation of the DAG's TaskGroup and its children
    used to construct the Graph and Grid views.
    """
    query = (
        session.query(
            TaskInstance.task_id,