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Hmm, my understanding was theories does not need to be falsifiable (e.g part of science) to be useful. CR is not falsifiable, it is not science. But not only science is useful. Am I wrong here?

That is, falsifiability was for me valid as a criterion of demarcation. But not only science is useful, that is positivism

Yes, my point was that there are theories in science that aren't falsifiable (it isn't a strict criterion of demarcation). More in this thread:https://twitter.com/reasonisfun/status/980084215571271680?s=21 …

Ok yes, I think I get it. In another wording, science is a special case of problem solving which involves critiques based on the outcome of experiments(though it is only a miniscule part of the critique). It is not a sharp line.

Would you agree to this phrasing: "Not every part of a scientific theory needs to be falsifiable, that is, needs to be criticizable via experiments?"

Yes, not just doesn't need but it would be impossible/bad — scientific ideas contain conjectures about abstract things! E.g. arguably, something that holds universally is not directly testable.