I agree with everything you say! But I would still argue that the things you describe (prediction of the partially known based on limited information) is built

I agree with everything you say! But I would still argue that the things you describe (prediction of the partially known based on limited information) is built on probability theory (sort of applying it in reverse), which in turn is based on assigning cases to reference classes and treating them as exchangeable in the ways that matter for the analysis at hand.