You can explain density as mass with arbitrarily small steps, yes. The Gaussian is the limit of many small binomials. With some graphical demonstrations to go a

You can explain density as mass with arbitrarily small steps, yes. The Gaussian is the limit of many small binomials. With some graphical demonstrations to go along with it, I've had much success getting people to fruitfully use these concepts. You also don't need a Jacobian, just the general concept of "How much does this change when we adjust this parameter in this direction". I'm not saying it's easier without calculus. By virtue of being more knowledge, the ability to use calculus will always be easier than not having it, all else equal. My argument is about opportunity cost. In my experience, people can get really good intuitions and tools around probability and statistics in the same time that they would otherwise learn the fine details around integrals and Jacobians -- and they'll find much more use for it.