For students who aren't going to learn calculus, I wonder if a better introduction to prob and stats (and perhaps math in general) would be through simulation and visualization rather than learning formulas by rote.

I had the opposite problem, which was that I learned stats as a bunch of proofs with practically no applications. I only use fairly basic stats today, but I often test my conclusions by running some sort of monte carlo analysis. I graph everything.

It sounds like a lot of what you're recommending could be described as teaching general quantitative scientific methodology.