Stats is a non-starter without calculus. You'd just be doing the "ram a bunch of rules for derivation and integration into your head." method since first principles require calculus. Linear algebra is less reliant on calculus, but the fun stuff still requires calculus. I learned basic linear algebra in high school for what it's worth, just enough to solve systems of linear equations.
Teach it the way you suggest and it would just be another useless math class that people would tune out of instead of grounding it in their everyday life experiences.
("Deriving from first principles" isn't the interesting stuff about calculus, either. And we're not requiring quantum physics to derive gravity from first principles in high school...)
I use linear algebra every day in my work as a graphics programmer. A huge portion of what I do depends on it. While I frequently need calculus it's still a minority of the time.
Derivates obviously show up everywhere: velocity, acceleration, gradients, ...
But it seems to me only the narrow portion relating to physically-based rendering or simulation requires an understanding of integrals. At the moment I can mainly think of how it's used to define continuous functions before introducing the discretization used to actually compute them.