Why are mole ratios central to solving stoichiometry problems?
You might as well ask why are addition and subtraction central to accounting.
All chemical reactions conserve (i) mass, and (ii) charge. It is convenient when we consider the stoichiometry of a chemical reaction to consider the mass in terms of moles, which necessarily represents a number of reactant and product particles, and thus also a mass of same. And this is why educators go to such lengths in teaching stoichiometry. If the reaction ain’t balanced with respect to mass and charge, it ain’t a model of reality. There should be many answers here that deal with stoichiometry.