Like that the digits of pi look a lot like the output of a random number generator? I don't think that prediction is seriously in dispute.

Yes, it is: https://en.wikipedia.org/wiki/Normal_number

But generating yet more empirical data that it still seems pretty random out to digit n does nothing to advance us to a proof that pi is normal.

I'm not sure about that. Maybe there is a very unlikely pattern somewhere around the n-th digit?

You could find a sequence of 1000 zeroes in the decimal expansion of pi and it would do nothing to prove or refute the idea that pi is normal.