Why does l'Ecuyer reject p-values close to 1? When applying several tests to a given generator, p-values smaller
than 0.01 or larger than 0.99 are often obtained by chance even if the
RNG behaves correctly with respect to these tests (such values should
normally appear approximately 2% of the time). In this case,
suspicious values would not reappear systematically (unless we are
extremely unlucky).
(Source,
page 6.)

Here's result where the test MaxOft AD produces a p-value close to 1 and so the RNG fails that test.
========= Summary results of SmallCrush =========
[...]
The following tests gave p-values outside [0.001, 0.9990]:
[...]
Test p-value
----------------------------------------------
[...]
6 MaxOft AD 1 - 3.6e-06
[...]
----------------------------------------------
All other tests were passed
[...]

NIST SP 800-22 calls such values &quot;perfect randomness&quot; (page 1-4, that is, PDF-page 16).

If a P-value for a test is determined to be equal to 1, then the
sequence appears to have perfect randomness. A P-value of zero
indicates that the sequence appears to be completely non-random.

So, what's wrong with a p-value close to 1?
So, what's wrong with a p-value close to 1?

Because random sequences don't look 'perfectly random' (or, rather, they have a low probability of doing so).  We're checking if this sequence quacks [1] like a perfectly random sequence; acting 'too uniform' is evidence that it is not.

[1]: from the proverb &quot;if it acts like a duck, and it quacks like a duck, then it is a duck&quot;.  Fairly common where I am; however you never know how regional sayings like this are..