Use airflow client in e2e tests

### Body

We have e2e tests, but it uses core airflow endpoints. potentially we could leverage airflow client , so that we have advantage of testing airflow client and also the real dags.

### Committer

- [x] I acknowledge that I am a maintainer/committer of the Apache Airflow project.

Start by locating the existing end-to-end tests and reviewing how they currently call core Airflow endpoints. Compare those calls with the Airflow client and determine which tests can exercise real DAGs through it; done means the relevant end-to-end coverage uses the client while still validating the real DAG behavior.

Written by the indexing model from the issue text.