Where angle measures and sines/cosines start to be useful is with uniform circular motion, which as you say is a calculus problem.
The key to understanding the form of trigonometry courses is to understand the context for their creation. Namely, all calculations (e.g. for astronomy, navigation, engineering, mapmaking, ...) used to be done by hand by humans, which was very expensive. It was important to have someone with fluent knowledge of trigonometric identities simplify formulas to a form with as few arithmetic operations and table lookups as possible to save money, or just to match the available function tables. Today in a computer age, extensive memorization of trigonometric identities is an anachronism, and spending lots of time on practicing their manipulation is okay algebra practice but not anything directly useful per se.