As an example, I'll run through a sample "scientifically pragmatic" argument against the death penalty: The death penalty is a terrible idea because it fails a cost benefit analysis. The benefit is a deterence to commit certain types crime, and a cost savings compared to life imprisonment. These are measurably small (research has shown that severity of a penalty has a nonlinear relationship with deterrent effect). The cost is the violation of a moral imperative not to kill. A society with the death penalty has made a conscious decision to compromise a moral imperative. This weakens the authority of moral imperatives, particularly the one against killing. The purpose of such moral imperatives is to promote a more civil, less violent society, because such societies enjoy an evolutionary advantage, hence their evolution in the first place.

How the argument proceeds from this point depends on core axioms - whether one values happiness or survival more, for instance. But the point is that it's possible to logically break down what appear to be moral questions into purely pragmatic ones, in service of deeper axioms.

The problem with this isn't that it's ineffective - it's that it's complicated, error prone, and relies heavily on a correct accounting of second-order effects and beyond. Science is hard - you need research and facts. Moral judgement is easy - you just say what pops into your head. As such, people mistrust the very idea that morals could be calculable, because it removes their agency. If someone has a strong, irrational feeling that Billy Murderer has it coming and should fry for what he did to those kids, my above argument (fleshed out properly to appeal to their core values, provided that's possible) will not convince them that it's a bad idea in the long run, no matter how scientific or testable it is. So they say things like "science has nothing to say". In fact, it does - we just don't want to listen.