> But so is a statistical test, or any 'nice' property that makes some algorithm behave better.

Not really. In general, the statistical tests are chosen such that when the law of large numbers comes into force, a perfectly random sequence would pass them with flying colors. If repeated runs of a perfect random number generator were failing these with a frequency that were improbable (i.e. far outside what the central limit theorem would predict), I think it would be safe to say the universe was playing a trick on you. Many of these are very closely related to predictability in practice as well. Uniformity is a statistical test, and I think it's probably the lowest bar you can set for "unpredictable".

The fundamental issue with measuring "randomness", especially for a PRNG, is that there is no set definition of information entropy suitable for calculation. Shannon entropy requires you to have a (possibly numerically sampled) PDF. The problem with that is that is you want to find the entropy of a sequence, you can't use the PDF of all the bytes you've gotten out the generator, since that PDF will have 1 for the exact sequence you obtained, and zero everywhere else. If you do it by multiple runs, there is no way you'd get enough data to construct an accurate PDF with a suitably large sample size. You have to use something based on highly subdivided sequence of values, but if you only look at each value in isolation you will only determine the uniformity with some small bin size, which ignores patterns due to proximity of values in the output.

So generally entropy is calculated using a more complex probability kernel, like the ones used in these statistical tests. But none of these can possibly be perfect. The "perfect" measure of entropy is Kolmogorov complexity[1], which captures the minimum length of a program you would need to be able to print the given output. On top of being undecidable in general, and also unobtainable in practice, you can show an upper bound of the Kolmogorov complexity on a CSPRNG that is actually quite low. Just take the algorithm the CSPRNG uses, plus any external entropy it got (from hardware interrupts, etc.). So in the end, various kinds of statistical tests are the only ones that make sense.

The actual goal of a CSPRNG is far more specific than just simulating a truly random sequence, which is why it differs from those used in simulations. It is unpredictability, defined by trying to make it so this short program which can generate all the output is impossibly hard to obtain from just the output in practice. In a vague sense, it is like (some of) the desired properties of a hash or cipher; we can't use a perfect random source (resp. one time pad), so let's get something that will take an impossible amount of computational power/measured output to break (resp. ciphertext using the same key), and mix it in with some as true random as we can get seed (resp. IV), and then mix some more in from time to time to make it even harder (resp. changing keys). That is the reason stream ciphers, while generally not as good as /dev/urandom-style CSPRNGs, are actually closely related enough that I think it's fair to call them CSPRNGs as well. They're just ones with a far worse algorithm/entropy source.

[1]: https://en.wikipedia.org/wiki/Kolmogorov_complexity