### 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)

"""Pool link rendering."""
        pool_id = self.get("pool")
        if pool_id is not None:
            url = url_for("TaskInstanceModelView.list", _flt_3_pool=pool_id)
            return Markup("<a href='{url}'>{pool_id}</a>").format(url=url, pool_id=pool_id)
        else:
            return Markup('<span class="label label-danger">Invalid</span>')
    def frunning_slots(self):
        """Running slots rendering."""
        pool_id = self.get("pool")
        running_slots = self.get("running_slots")
        if pool_id is not None and running_slots is not None:
            url = url_for("TaskInstanceModelView.list", _flt_3_pool=pool_id, _flt_3_state="running")
            return Markup("<a href='{url}'>{running_slots}</a>").format(url=url, running_slots=running_slots)
        else:
            return Markup('<span class="label label-danger">Invalid</span>')
    def fqueued_slots(self):
        """Queued slots rendering."""
        pool_id = self.get("pool")
        queued_slots = self.get("queued_slots")
        if pool_id is not None and queued_slots is not None:
            url = url_for("TaskInstanceModelView.list", _flt_3_pool=pool_id, _flt_3_state="queued")
            return Markup("<a href='{url}'>{queued_slots}</a>").format(url=url, queued_slots=queued_slots)
        else:
            return Markup('<span class="label label-danger">Invalid</span>')
    def fscheduled_slots(self):