For reference, I never took "freshman calculus" and never did Taylor Series at University, and I have a PhD in discrete math. I do know about TS, but it was axtra-curricula.
In all fairness, I never took "freshman calculus" either; that said, most undergraduate math degrees require either real or complex analysis, and at the graduate programs with which I am familiar you would have a very difficult time passing the qualifying exams if you weren't comfortable working with Taylor series.
I've gone and checked. They were mentioned, effectively with "when you need them they will be pretty much obvious" - and they were. They never formed a topic in their own right, they just "fell out" of the actual material.
This was in Australia in the early 80s, and to this day I think of TSs as sort of obvious. But I did Pure Maths, and I think TSs are more emphasised in applied where you use them for approximations, etc.
It's way too long ago to have any measure of that now. I did my 4 year bachelor's degree from 1980 to 1982 inclusive, taking nearly 4.5 years worth of courses in that time. I did Analysis I, Analysis II, Complex Analysis and Mathematical Methods all in my first chronological year, and Taylor Series were never actually a subject of discussion. My recollection is that they were defined, and it was just pretty obvious that they worked when everything was "nice", and that things could and would go wrong if things were in any way "odd".
But this was 32 years ago, and while I've used calculus in one form or another constantly, I've never had to do anything "interesting" with Taylor Series. Now looking back I'm hard pressed to say what was easy and what wasn't, what was obvious and what wasn't.
It would actually be interesting to go be an undergraduate and see what is obvious now with the years under my belt, and what needs to be thought about hard.
I suspect the fact my degree was almost as long ago, and I certainly _can_ remember the real analysis topics that were non-obvious, means there were fewer of them for you than for me :-)
Or that your course was harder than mine, or that I was struggling with more advanced courses at the same time, and therefore had a different perspective on it all.
