### 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)

x=x_points[task_id],
                y=scale_time_units(y_points[task_id], y_unit),
            )
        max_date = max(ti.execution_date for ti in tis) if tis else None
        session.commit()
        form = DateTimeWithNumRunsForm(
            data={
                "base_date": max_date or timezone.utcnow(),
                "num_runs": num_runs,
            }
        )
        chart.buildcontent()
        return self.render_template(
            "airflow/chart.html",
            dag=dag,
            chart=Markup(chart.htmlcontent),
            height=str(chart_height + 100) + "px",
            root=root,
            form=form,
            tab_title="Landing times",
            dag_model=dag_model,
        )
    @expose("/paused", methods=["POST"])
    @auth.has_access(
        [
            (permissions.ACTION_CAN_EDIT, permissions.RESOURCE_DAG),
        ]
    )
    @action_logging
    def paused(self):