Correct. The value judgements (feelings) of the demos should be reflected in the policies that the democracy enacts and is governed by. That's part of what a government of and by the people means. Governing by any other system, like utilitarianism, or arbitrary moral principles like 'killing criminals is wrong' that the majority of constituents don't hold, needs heavy justification, and may not be justifiable in a democracy. If you'd prefer an oligarchy or monarchy wherein someone can impose their own minority value system on the people, even if that system objectively results in 'better outcomes' whatever that means to you, that's fine, but just be honest about it.

In brief, it doesn't matter what "is best" in some utilitarian framework if the democracy is not attuned to that framework. It doesn't matter if the death penalty actually increases crime. What matters is that the government should be doing what the people want it to be doing. If the people really want to kill criminals, any system that prevents that is non-democratic.