I just discovered the repo that contains end-to-end tests for Open edX - https://github.com/openedx/end-to-end-tests, which is only ~2 months old. In the past, I believe this was something that we wanted, although I don’t exactly remember what for. The README states that the suite is deployment-agnostic, meaning that we could run these tests as part of our instance test checklist on the instances that we deploy on our clusters.
Either way, just wanted to share this to raise awareness, in case someone comes up with a good way to integrate them into our work.