As far as I can tell, Microsoft has not ported the no-code capabilities in Azure Machine Learning Studio from v1 to v2. If you go to the "Automated ML" authoring page and click “+ New Automated ML job”, it uses v1, and there’s no way to tell it to use v2. Unless Microsoft decides to migrate it to v2, it's likely that the "Automated ML" authoring page will become increasingly unusable over time (as we've already seen with the testing function no longer working). Something similar happened with the no-code capabilities of the "Designer" authoring page. It's no longer possible to run an end-to-end experiment using Designer's no-code components due to incompatibilities with v2.
In summary, I think the answer to my original question is that Microsoft decided not to migrate the no-code capabilities in Azure Machine Learning Studio from v1 to v2 of the API, so those capabilities will continue to degrade over time.