The differences here compared to unit tests are that the breaking is outside of your control and the updating process is tedious. Also, testing requires making real API calls rather than using stubs so it requires additional infrastructure.
You are right that fine tuning would probably help to minimize the risks, but it probably never can be zero. New tests will also be needed when customers find new edge cases that break our assumptions.
Testing LLM prompts is a new paradigm that we'll have to learn to deal with.