It's difficult to map each of the 15 tests to a particular qualitative descriptor. Adjectives like robust are not mentioned in the 800-22 documentation.
The null hypothesis of randomness ($H_0$) in all tests fundamentally requires samples from a stationary and ergodic process. Otherwise the tests would be inconsistent with respect to time, and could not accurately model the samples' shifting distributions or test score expectations. Or the test scores for this million bits would differ widely from that million's. Although be mindful of some variability implicit in any random process.
The requirement for independence within each test varies though. §2 of 800-22 details the mathematical assumptions underpinning each test. NIST doesn't always highlight the level of significance of independence within each test. They often don't attempt to debate the causality of a $H_0$ sequence, just it's effect and characteristics.
Some are explicitly discussed. The Discrete Fourier Transform (Specral) test is an example of a test that specifically targets independence in the mathematical sense. It looks for correlations between samples spaced $n$ samples apart. Maurer’s "Universal Statistical" test is similar, exploiting correlation for compressibility. As is the Binary Matrix Rank test. Conversely the Frequency (Monobit) test doesn't. It will accept any degree of correlation as long as the ones and zeros are evenly matched in line with a half normal distribution.
The sensitivity to independence is harder to identify with some of the other tests. You may have to decompose the statistics beneath tests like the Serial or Cumulative Sums (Cusum) tests.
Don't confuse the recommended input lengths with non ergodicity. The recommendations are there simply to facilitate some of the test metrics' reference distributions, or to limit processing time.