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.

Obviously you’re right. But we are talking kids here. There is so much to learn in statistics at basic to intermediate level without algebra. Many social scientists or in the modern age data scientist go through life with cookbook statistics and some code examples. Sure, it lacks rigour, but intuition especially for kids breeds interest and then the subset that is mathematically inclined can go on to do statistics with all rigour.

I agree. Algebra based Statistics is one of the things that makes people say "Math is a religion". There is a lot of "just trust me" in statistics when you don't know calculus.

I suppose you could switch Probability & Statistics to "Just Enough Calculus" & Statistics to make it a less religious experience... but that also sort of defeats the "nobody really uses Calculus" mentality.