Agreed with your overall point, and your specific example. Myself and one of my good friends I met in my physics classes in college both felt the importance of Taylor series had been massively undersold in our calculus courses, because it just kept coming up in our various physics courses. I learned it just as a thing that existed, but we kept relying on them when deriving things in courses like thermal dynamics or mechanics. We would joke that calculus professors should stop the class and just emphasize, "This is really important!" But of course, that wouldn't make the material land any better, for the reasons you've explained.