Agreed, there is definitely a huge gap between doing, for example, a t-test and deriving the t-test (or giving any substantial justification for its form). You need a class in analysis, a class in measure-theoretic probability, then a mathematical statistics class to bring you back to the topic. It's a tough square to circle.

Most who teach stats at the undergrad or hs level have little choice but to defer a lot of things to a later course.

You just use discrete scenarios, not continuous. Same concepts, easier math.

same way you can do linear geometry without calculus.

In my experience, just knowing discrete probability and statistics is profoundly unuseful, especially when you walk out of class thinking you know the subject, sign up for a next course that involves real-valued quantities, and bomb everything.