Earlier quoted context omitted.
"Independently" is also a loaded word - one could mean that there's no direct causal connection between the state of the system and the state of the measurement apparatus, or one could mean that those two states are (in ensemble experiments) independent in the statistical sense. In order for Bell's theorem (and a lot of subsequent quantum theory) to work, we have to interpret causal and statistical independence as be…
"Bell, in particular, justified it by claiming that it was unscientific to assume that experimenters didn't have free will. Okay, sure." You make some good points. I am not sure how Bell meant this, but read out of context, this is obviously false. Deterministic modelling (and therefore lack of absolute free will of the components of the system) is a long tradition of science.
Regarding the context, the quote I'm thinking of comes from one of the essays in Speakable and Unspeakable in Quantum Mechanics, which is an enlightening read, both in terms of content and for its historical value. The issue, I think, is that the kind of 'conspiracy' implied by superdeterminist theories, wherein nature sort of guides our hand in picking certain experimental settings so as to hide information seems troubling to many scientists. See Zeilinger's quote at https://en.wikipedia.org/wiki/Superdeterminism