I don't think you're correct that the z-score is not being interpreted correctly. I haven't been able to find research demonstrating any non-randomness in pi, so without that we have nothing to compare it to.
There's a large literature of people looking for non-randomness in pi, and failing.
If you try 50,000 different tests, with a z-score of 3.29 then 0.1% - 50 of them - will give false positives.