Do hall effect-based current sensors (clamps) always use a semiconductor as the material through which we apply the initial voltage difference and why?
Although you get the Hall effect in metals as well, the size of the Hall voltage is inversely proportional to the number of mobile charge carriers. Metals typically have an electron or two per atom in the conduction band. Semiconductors have only one mobile carrier per dopant atom, which is orders of magnitude fewer.
The Hall voltage in semiconductors is large enough to be useful, and have a decent signal to noise ratio when amplified. The Hall voltage in metals is much more difficult to measure.