Is that correct to say that semantic consequence is equivalent to the concept of sound argument in classical propositional logic? If it is the case, arguments or theories with contradictory premises are of no interest of study. If it is not the case, what is the rationale to lead with arguments and theories with contradictory premises if everything can be proved either true or false?
In the way the terms are standardly used, logical or semantic consequence, validity, and entailment are interdefinable. An argument is (deductively) valid if and only if its conclusion is the consequence of its premises and if and only if the premises entail the conclusion. An argument is sound if it is valid and has true premises. So semantic consequence corresponds to validity, not soundness.
As you say, in classical propositional logic an argument with contradictory premises is of no interest, since it entails everything and can never be sound. There are non-classical logics that do not have this property: the term for these is paraconsistent.