### 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)

"""List loaded plugins."""
        plugins_manager.ensure_plugins_loaded()
        plugins_manager.integrate_executor_plugins()
        plugins_manager.initialize_extra_operators_links_plugins()
        plugins_manager.initialize_web_ui_plugins()
        plugins = []
        for plugin_no, plugin in enumerate(plugins_manager.plugins, 1):
            plugin_data = {
                "plugin_no": plugin_no,
                "plugin_name": plugin.name,
                "attrs": {},
            }
            for attr_name in self.plugins_attributes_to_dump:
                attr_value = getattr(plugin, attr_name)
                plugin_data["attrs"][attr_name] = attr_value
            plugins.append(plugin_data)
        title = "Airflow Plugins"
        doc_url = get_docs_url("plugins.html")
        return self.render_template(
            "airflow/plugin.html",
            plugins=plugins,
            title=title,
            doc_url=doc_url,
        )
class ProviderView(AirflowBaseView):