How are 8-bit versions of "such things" expected to perform on
statistical tests for randomness?

Exactly as specified in the standard? For generators, I'm pretty sure it's specified mathematically what they produce with perfect clarity.

For distributions, I don't believe we have such a precise specification. You could look at the existing implementations in libc++ and libstdc++ if you really wanted to know. I don't see any innate reason why they would be of lower or greater statistical quality than the distributions for other unsigned integers.