I think a "statistics" class could still help a lot of people without ever touching continuous distributions (which as you mention are more naturally built off calculus) or multivariate distributions (which as you mention are more naturally built off linear algebra). Simple stuff like basic conditional probability, Bayes' rule, confidence intervals, basic hypothesis testing, and general "statistical thinking" don't really require calculus or linear algebra. Just getting people to a point where they understand some of the really basic tools seems pretty valuable, and pretty far from the current state of things.