### 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)

"""Set state to running."""
        return self._set_dag_runs_to_active_state(drs, State.RUNNING)
    @provide_session
    def _set_dag_runs_to_active_state(self, drs: list[DagRun], state: str, session: Session = NEW_SESSION):
        """This routine only supports Running and Queued state."""
        try:
            count = 0
            for dr in session.query(DagRun).filter(DagRun.id.in_([dagrun.id for dagrun in drs])):
                count += 1
                if state == State.RUNNING:
                    dr.start_date = timezone.utcnow()
                dr.state = state
            session.commit()
            flash(f"{count} dag runs were set to {state}.")
        except Exception as ex:
            flash(str(ex), "error")
            flash("Failed to set state", "error")
        return redirect(self.get_default_url())
    @action(
        "set_failed",
        "Set state to 'failed'",
        "All running task instances would also be marked as failed, are you sure?",
        single=False,
    )
    @action_has_dag_edit_access
    @provide_session
    @action_logging
    def action_set_failed(self, drs: list[DagRun], session: Session = NEW_SESSION):