Run tests for start and finish task also with private cert

[michaelsauter]

Before #722, it was possible to run all tests using a private cert, by supplying -private-cert to go test. The downside of this was that all tests on GitHub ran without it, to avoid overly long test times. Now that the overall test time is way shorter, we can afford to add one test for each task to run with private cert enabled.

Tasks:

• Updated design documents in docs/design directory or not applicable
• Updated user-facing documentation in docs directory or not applicable
• Ran tests (e.g. make test) or not applicable
• Updated changelog or not applicable