### Apache Airflow version

2.6.0

### What happened

Ever since we upgraded to Airflow 2.6.0 from 2.5.2, we have seen that there is a warning stating "The scheduler does not appear to be running" intermittently.

This warning goes away by simply refreshing the page. And this conforms with our findings that the scheduler has not been down at all, at any point. By calling the /health point constantly, we can get it to show an "unhealthy" status:

These are just approx. 6 seconds apart: 

```
{"metadatabase": {"status": "healthy"}, "scheduler": {"latest_scheduler_heartbeat": "2023-05-11T07:42:36.857007+00:00", "status": "healthy"}}

{"metadatabase": {"status": "healthy"}, "scheduler": {"latest_scheduler_heartbeat": "2023-05-11T07:42:42.409344+00:00", "status": "unhealthy"}}
```

This causes no operational issues, but it is misleading for end-users. What could be causing this? 

### What you think should happen instead

The warning should not be shown unless the last heartbeat was at least 30 sec earlier (default config).  

### How to reproduce

There are no concrete steps to reproduce it, but the warning appears in the UI after a few seconds of browsing around, or simply refresh the /health endpoint constantly.

### Operating System

Debian GNU/Linux 11

### Versions of Apache Airflow Providers

apache-airflow-providers-amazon==8.0.0
apache-airflow-providers-celery==3.1.0
apache-airflow-providers-cncf-kubernetes==6.1.0
apache-airflow-providers-common-sql==1.4.0
apache-airflow-providers-docker==3.6.0
apache-airflow-providers-elasticsearch==4.4.0
apache-airflow-providers-ftp==3.3.1
apache-airflow-providers-google==10.0.0
apache-airflow-providers-grpc==3.1.0
apache-airflow-providers-hashicorp==3.3.1
apache-airflow-providers-http==4.3.0
apache-airflow-providers-imap==3.1.1
apache-airflow-providers-microsoft-azure==6.0.0
apache-airflow-providers-microsoft-mssql==3.3.2
apache-airflow-providers-microsoft-psrp==2.2.0
apache-airflow-providers-microsoft-winrm==3.0.0
apache-airflow-providers-mysql==5.0.0
apache-airflow-providers-odbc==3.2.1
apache-airflow-providers-oracle==3.0.0
apache-airflow-providers-postgres==5.4.0
apache-airflow-providers-redis==3.1.0
apache-airflow-providers-sendgrid==3.1.0
apache-airflow-providers-sftp==4.2.4
apache-airflow-providers-slack==7.2.0
apache-airflow-providers-snowflake==4.0.5
apache-airflow-providers-sqlite==3.3.2
apache-airflow-providers-ssh==3.6.0


### Deployment

Official Apache Airflow Helm Chart

### Deployment details

Deployed on AKS with helm

### Anything else

None more than in the description above.

### 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)

"""
        This function is where the main scheduling decisions take places.
        It:
        - Creates any necessary DAG runs by examining the next_dagrun_create_after column of DagModel
          Since creating Dag Runs is a relatively time consuming process, we select only 10 dags by default
          (configurable via ``scheduler.max_dagruns_to_create_per_loop`` setting) - putting this higher will
          mean one scheduler could spend a chunk of time creating dag runs, and not ever get around to
          scheduling tasks.
        - Finds the "next n oldest" running DAG Runs to examine for scheduling (n=20 by default, configurable
          via ``scheduler.max_dagruns_per_loop_to_schedule`` config setting) and tries to progress state (TIs
          to SCHEDULED, or DagRuns to SUCCESS/FAILURE etc)
          By "next oldest", we mean hasn't been examined/scheduled in the most time.
          We don't select all dagruns at once, because the rows are selected with row locks, meaning
          that only one scheduler can "process them", even it is waiting behind other dags. Increasing this
          limit will allow more throughput for smaller DAGs but will likely slow down throughput for larger
          (>500 tasks.) DAGs
        - Then, via a Critical Section (locking the rows of the Pool model) we queue tasks, and then send them
          to the executor.
          See docs of _critical_section_enqueue_task_instances for more.
        :return: Number of TIs enqueued in this iteration
        """
        
        with prohibit_commit(session) as guard:
            if settings.USE_JOB_SCHEDULE:
                self._create_dagruns_for_dags(guard, session)
            self._start_queued_dagruns(session)
            guard.commit()
            dag_runs = self._get_next_dagruns_to_examine(DagRunState.RUNNING, session)