I am in favor of random tests, and I write them. However, whether they are appropriate in a particular build environment and which test suites they should be included in is a more nuanced question.
Run locally (e.g., overnight on your dev box) randomized tests have found bugs both obvious and obscure. The obscure ones are arcane enough that I think random testing was really the only realistic one to flush them out. As a test, I took one tough-to-find bug discovered via randomized testing and had a half dozen crack developers review the function (about a dozen lines of code) where it occurred. None were able to detect it.
Many of your arguments against randomized data are flavors of "the test isn't reproducible". However, a well written randomized test will capture the seed used to start the randomized seed and output it on failure. In addition to allowing you to repeat the test by hand, this allows you to trivially create new test which test the specific issue by hardcoding the seed for that test. Of course, it's probably nicer to hand-code an explicit test covering that case, but laziness has its virtues, and this even allows you to essentially auto-generate new test cases from a failing seed.
The one point you make that I can't debate, however, is that it breaks the build systems. Most build and continuous integration tests expect the tests to do the same thing, every time. So a test that randomly fails will create chaos, randomly failing and pointing the fingers at changes that were harmless.
A solution then, is to still run your randomized tests as part of the build and CI tests, but run it with a fixed seed, for a fixed number of iterations. Hence the test always does the same thing, but still explores a bunch of the input space (if you run it for multiple iterations).
Locally, e.g., when changing the concerned class, you are free to run it for more iterations or with other seeds. If randomized testing ever becomes more popular, you could even imagine a specific suite of tests which are known to be random, which could be run with different seeds (hence with increasing coverage over time), and where failures wouldn't mean the same thing as deterministic CI systems (i.e., runs aren't associated 1:1 with code changes and so you don't point a finger at a particular change when things fail).
There is a lot to be said for randomized tests, especially well written ones, so don't be too quick to dismiss them!