I'm not sure who told you that non-random tests means non-parameterized tests but I don't think you'll get a lot of push back on parameterized tests from any of

I'm not sure who told you that non-random tests means non-parameterized tests but I don't think you'll get a lot of push back on parameterized tests from any of the commentators here. The point is you should write exhaustive tests such that you couldn't imagine the randomized test finding anything new, especially on bounded sets of inputs. If you're not writing exhaustive tests because of a random strategy then yes, the choice is exactly between knowing immediately or later. I've seen novices assume random tests are exhaustive even when they could think of several edge cases.