This has been known and lamented for a century, and it long ago transitioned from being a radical idea to being conventional wisdom. It has been orthodoxy for decades (although recently there has been some reaction against the orthodoxy by a few experimental charter schools.) So, if current math education involves a lot of rote learning, it's not for lack of awareness or lack of attempts to fix the problem. In rebellion against facts and rote learning, teachers have elevated understanding above problem-solving as the highest goal of mathematics education, but it turns out to be impossible to demonstrate understanding, much less engage real-world problems, without knowing a lot of handy facts. That's why the most idealistic, anti-rote-learning teachers still want their kids to learn that sin^2(x) + cos^2(x) = 1. They can't demonstrate the real-world relevance of mathematics or help kids understand abstract concepts unless the kids have some basic problem-solving competence that doesn't involve spending hours searching through the book for fundamental facts.

And yes, they do want kids to get to the point where they can "forget" that sin^2(x) + cos^2(x) = 1 and regenerate that knowledge from visualizing the unit circle and applying the Pythagorean Theorem. That ideal has also been orthodoxy for decades. It just turns out that in practice, deeper understanding emerges from practice, and practice requires competence. Rote learning is a way of bootstrapping competence so competence can be turned into understanding. Believe me, progressive, "kinder and gentler" teachers were the norm in all the schools I've gone to, and they all tried every trick they knew to help us skip straight from ignorance to deep understanding with as little dry, repetitive work as possible, but they could not get around the need to learn many facts and techniques by heart.

By the way, the route through calculus you describe reflects that you are basically the kind of kid that the schools don't worry about. I was the same kind of student. It might seem strange to us, but to most students, the history of a subject and the problems that originally motivated it are the very definition of dry and disconnected. Nobody gives a damn about the awesome intellectual journeys of some dead nerds except kids like you and me who will learn the subject regardless of how it's taught. Differentiation is introduced first precisely because it makes the subject less dry to most students, because it allows them to immediately apply what they learn to simple concrete problems involving rates.