We don't have those integration tests. They happened by chance when customers did things. Do we really have tests if we cannot define those tests, if we cannot run those tests? If we make a significant change that could feasibly affect the outcome of those tests, and we are unable to run those tests, do we have those tests?

"How do you "design a good system" without testing it?"

We must be coming at this through such widly different contexts. To me, it is simply obvious and normal that it's possible to create a good design for something, and that good design can exist before any tests have ever been created or executed.

To me, that you asks that question suggests that we have such different contexts that we might as well be speaking different languages. I would be horrified that people would churn out a rubbish design and just let tests handle all the crap and force it into a good design; but I do gather that's normal procedure in some industries.