The design approach of just iterating until it passes a suite of statistical tests is ... not the best.
The issue is that the common suites of statistical test don't include every credible statistical test at their level of complexity.
Essentially as you retry against the tests you expend their utility, producing a result that is guaranteed to pass them as a byproduct of your process. ... but may in fact be extraordinarily weak to yet another reasonable statistical test (and/or broken in some practical application) that no one has yet thought to include in the test suite.
The issue is that the common suites of statistical test don't include every credible statistical test at their level of complexity.
Essentially as you retry against the tests you expend their utility, producing a result that is guaranteed to pass them as a byproduct of your process. ... but may in fact be extraordinarily weak to yet another reasonable statistical test (and/or broken in some practical application) that no one has yet thought to include in the test suite.