A few good points have already been made in the other responses, and I want to add an additional observation: To me, the core of the issue is not in choosing between "non-deterministic" testing vs. a tiny set of hand-chosen examples.
Rather, to me, the issue is exhaustive testing vs. non-exhaustive testing. Typically, critical issues in complex systems only arise extremely rarely. Consequently, randomized testing typically does not bring them to light. There may definitely be cases where it is useful, and it can certainly easily be more exhaustive than a small hand-picked set of unit tests. But the core issue is still that I typically need exhaustive tests to be more certain about relevant properties.
Rather, to me, the issue is exhaustive testing vs. non-exhaustive testing. Typically, critical issues in complex systems only arise extremely rarely. Consequently, randomized testing typically does not bring them to light. There may definitely be cases where it is useful, and it can certainly easily be more exhaustive than a small hand-picked set of unit tests. But the core issue is still that I typically need exhaustive tests to be more certain about relevant properties.