Restructure integration tests

Our integration tests currently take a substantial amount of time to run, and due to their setup they're essentially forced to run synchronously instead of parallel. @Gankra has suggested we restructure the tests so that we have fewer of them overall, and so that a given test can exercise groups of features instead of just one thing. We should determine exactly what pairings of features we want to use and how many tests we want to end up with.

The issue names no files, tests, or entry points. Start by locating the integration-test suite and documenting its setup and feature coverage, then determine which feature pairings and final test count are appropriate. Done means the test structure and grouping are agreed, with fewer tests and a path toward parallel execution.

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