In the dieharder test suite there are things like:-
diehard_3dsphere| 3| 4000| 100|0.56011310| PASSED
and in the NIST IID (brief mode) and randomness test suites there are things like:-
** Passed IID permutation tests
and:-
1 2 1 1 0 2 2 0 0 1 0.739918 10/10 Universal
1 0 2 3 1 0 0 1 1 1 0.534146 10/10 ApproximateEntropy
0 1 2 0 2 1 0 1 0 0 ---- 7/7 RandomExcursions
2 0 0 1 3 0 0 0 0 1 ---- 7/7 RandomExcursions
0 0 0 1 0 1 1 1 2 1 ---- 7/7 RandomExcursions
2 0 0 0 1 2 0 2 0 0 ---- 7/7 RandomExcursions
0 1 1 0 0 3 0 1 1 0 ---- 7/7 RandomExcursions
0 0 0 2 0 1 1 0 2 1 ---- 7/7 RandomExcursions
0 0 1 0 1 1 0 1 2 1 ---- 7/7 RandomExcursions
1 0 1 0 2 2 1 0 0 0 ---- 6/7 RandomExcursions
0 0 2 0 2 1 0 1 1 0 ---- 7/7 RandomExcursionsVariant
0 1 1 1 0 2 1 0 0 1 ---- 7/7 RandomExcursionsVariant
0 2 1 0 1 0 1 1 1 0 ---- 7/7 RandomExcursionsVariant
0 2 1 0 1 0 0 0 2 1 ---- 7/7 RandomExcursionsVariant
0 1 1 0 0 2 0 0 2 1 ---- 7/7 RandomExcursionsVariant
0 0 1 1 0 0 1 1 2 1 ---- 7/7 RandomExcursionsVariant
0 0 0 1 2 1 0 1 2 0 ---- 7/7 RandomExcursionsVariant
0 0 1 1 0 2 1 0 1 1 ---- 7/7 RandomExcursionsVariant
0 1 0 2 1 0 0 0 2 1 ---- 7/7 RandomExcursionsVariant
0 2 0 1 1 1 0 0 2 0 ---- 7/7 RandomExcursionsVariant
0 1 1 1 0 2 0 1 1 0 ---- 7/7 RandomExcursionsVariant
0 1 0 1 0 1 0 0 1 3 ---- 7/7 RandomExcursionsVariant
0 0 1 0 1 0 2 1 1 1 ---- 7/7 RandomExcursionsVariant
0 0 1 0 2 1 0 2 1 0 ---- 7/7 RandomExcursionsVariant
0 0 1 1 1 0 1 2 1 0 ---- 7/7 RandomExcursionsVariant
0 0 0 0 2 1 0 3 1 0 ---- 7/7 RandomExcursionsVariant
0 0 0 0 1 2 0 3 1 0 ---- 7/7 RandomExcursionsVariant
0 0 0 1 1 2 0 0 1 2 ---- 7/7 RandomExcursionsVariant
0 0 2 3 1 0 1 1 0 2 0.350485 10/10 Serial
1 2 1 1 1 1 1 0 1 1 0.991468 10/10 Serial
In the above you will notice that there are no p values available for the RandomExcursionsVariant tests (as the sample was only 10 MB). p values are uniformly distributed $ \mathcal{U}(0,1) $ so they fall randomly anywhere within that interval. They can be high, low or middling. And occasionally we even expect them to be outside of the critical $\alpha$ value for what we know to be valid data samples. IMHO, the failure or success of the these tests is most clearly conveyed by a test score (e.g. 7/7) as above, or a semantic message stating so, e.g. "PASS.."
Given that p values sometimes do not even appear, and are randomly distributed when they do, what value are they in randomness testing?