✅ Resurrect core integration tests

[JMicheli]

I'm starting this as a draft because I think there's work to do, but I want to see codecov's results for it.

This moves the contents of the old, unused core/integration-tests to the normal path for integration tests core/tests and adds a couple of actual integration tests.

I think the tests could be expanded and better-written. I tried to translate some other old tests but it seems the core API has changed enough that it wasn't obvious how they should be re-written.