On the other hand, some tests could be aligned enough with the needs of a specific application and prove that a PRNG is good, not only probably good. For examp

On the other hand, some tests could be aligned enough with the needs of a specific application and prove that a PRNG is good, not only probably good. For example, if a game uses a PRNG to "stretch" a random seed for procedural generation purposes conditional probabilities are important (to avoid spurious correlations between features) but period is completely irrelevant (even a relatively short period would be many orders of magnitude larger than output size).