Testing always helps, no doubts about that. But there are still some situations where testing is just not enough. No amount of unit testing can check the behavior of your code in a real production environment.

I would disagree with that completely. You likely won't have 100% test coverage on production systems but you should have all the core case covered by unit and integration tests. You should also have a some blackbox API level tests.

If you change some code and the tests are still passing, provided your test suite is well written, then the code should perform the way you expect in prod. If it doesn't then your test suite needs to be improved but the value of well written tests can't be under estimated.

I still fail to see how this would catch something a unit/integration/API test wouldn't.