### 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)

method_permission_name = {
        "list": "read",
    }
    base_permissions = [
        permissions.ACTION_CAN_READ,
        permissions.ACTION_CAN_ACCESS_MENU,
    ]
    list_columns = ["id", "dttm", "dag_id", "task_id", "event", "execution_date", "owner", "extra"]
    search_columns = ["dttm", "dag_id", "task_id", "event", "execution_date", "owner", "extra"]
    label_columns = {
        "execution_date": "Logical Date",
    }
    base_order = ("dttm", "desc")
    base_filters = [["dag_id", DagFilter, lambda: []]]
    formatters_columns = {
        "dttm": wwwutils.datetime_f("dttm"),
        "execution_date": wwwutils.datetime_f("execution_date"),
        "dag_id": wwwutils.dag_link,
    }
class TaskRescheduleModelView(AirflowModelView):
    """View to show records from Task Reschedule table."""
    route_base = "/taskreschedule"
    datamodel = AirflowModelView.CustomSQLAInterface(models.TaskReschedule)  
    related_views = [DagRunModelView]
    class_permission_name = permissions.RESOURCE_TASK_RESCHEDULE
    method_permission_name = {
        "list": "read",
    }
    base_permissions = [
        permissions.ACTION_CAN_READ,