I don't understand the statistics-vs-calculus debate at all. Probability theory is at the heart of statistics, and how can a student understand statements like 'the integral of a probability density function must equal one over its possible range' without taking some calculus?
The problem with calculus as taught today in most schools is that most of the mathematical tricks for integrating functions that they hammer the students with are essentially useless in real-world problems so you might as well just start with numerical methods of approximation right at the beginning, once basic concepts like 'the integral of a function corresponds to the area under the curve of that function' are grasped. I know many math purists are dedicated to the pencil and paper approach, but it's not all that useful for non-math majors who will certainly be building or using computational models almost exclusively.
Another prerequisite for statistics should be linear algebra, because without a grasp of vector representations of data and matrix operations on that data, a lot of statistical concepts, e.g. multivariate statistics, won't be very understandable.
Fundamentally, trying to use statistics without this deeper understanding of the mathematics underlying much of it can lead to all kinds of issues, like choosing the wrong statistical method for a given problem.
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.
A mandatory nitpick: you cannot do conditional probability without multivariate distributions, but you are right, you do not need linear algebra for a basic introduction. Even continuous distribution can be introduced, you just start with discrete cases, introduce areas in a histogram, and then approximate the histogram by a continuous curve. You can even get to the Riemann integral that way if you want to. And nobody actually calculates normal probabilities by integrating the Gaussian function, it's kind of hard to do.
The problem with calculus as taught today in most schools is that most of the mathematical tricks for integrating functions that they hammer the students with are essentially useless in real-world problems so you might as well just start with numerical methods of approximation right at the beginning, once basic concepts like 'the integral of a function corresponds to the area under the curve of that function' are grasped. I know many math purists are dedicated to the pencil and paper approach, but it's not all that useful for non-math majors who will certainly be building or using computational models almost exclusively.
Another prerequisite for statistics should be linear algebra, because without a grasp of vector representations of data and matrix operations on that data, a lot of statistical concepts, e.g. multivariate statistics, won't be very understandable.
Fundamentally, trying to use statistics without this deeper understanding of the mathematics underlying much of it can lead to all kinds of issues, like choosing the wrong statistical method for a given problem.