Maybe things were different back when OP took calc but I’d bet any standard calculus textbook used for instruction in the past 20 years mentions this many times.
Mine didn't lead with it. It may have been buried in there somewhere but it wasn't highlighted as critically important. Instead you just got the mechanics of differentiating and integrating functions thrown at you without explanation.
This was back in the late 1990s and early 2000s. I hope things have improved.
We had a very rigurous program in highschool back in Eastern Europe, it was so good that taking clac in college in the US was redundant. However, I remember having a similar click moment when I visually or intuitively understood derivatives and integrals. Before that it juggling math mechanics, which in itself is also not bad and helps build a certain muscle that can be used later on. But I am familiar with the fumbling a bit through some math classes.
My conclusion is different though. I think there are different types of thinkers and different teaching styles and when the subject becomes very complex the disparity between the two is exacerbated. And luck plays the role in matching up with the right teacher/professor for you.
It's unfortunate you were in a geographic location and a time that exposed you to poor math teachers.
I was taught calculus in high school, well before university, in Australia in the late 1970s.
Visual analogies such derivatives being tangential rates of change and 2nd derivatives being local curvatures were taught and drawn on the board in the first week of two years of Calc I and Calc II classes that came before leaving high school to attend university.
This was back in the late 1990s and early 2000s. I hope things have improved.