There seem to be two questions here.
Meaning that the formulas it took to derive Bell's inequality doesn't seems to have any indication of quantum effects, so it seems obvious that it would turn out not matching quantum effective results.
The setting used to derive Bell's inequalities is neither classical nor quantum.
Rather, it only assumes that measurement results are described by probability distributions. One then imposes some "natural" assumptions over these probability distributions and investigates the correlations that can arise from them.
The fact that measurement outcomes are to be described, in the most general case, by a probability distribution, is general enough to encompass any kind of theory, classical, quantum, or other.
Indeed, it is hard to imagine a physical theory in which this would not be the case.
But how was hidden variable (which seems classical) supposed to have been a solution to incompleteness in the first place?
Indeed, hidden variable theories are "classical". The reason why one wonders whether a kind of hidden variable theory can explain the predictions of quantum mechanics is that many don't feel very comfortable with the idea that nature really works like quantum mechanics seems to be telling us.
For example, the intrinsic indeterminism associated with measurements is completely at odds with the classical (deterministic) way of describing the world.
The fact that quantum mechanics predicts nonlocal correlations that however do not allow for superluminal communication can also be seen as quite odd.
It is therefore only natural that people wondered whether it is possible to go back to using a more "natural" and intuitive description of the world.
However, Bell's inequalities tell us that this is indeed not possible.