I disagree with you on trig. Trig is the study of straight lines and what you can do with them. The important part of calculus is the fact that you can approximate curves by straight lines [1]. A folk theorem in applied math is that 90% of the time, an approximation by straight lines is good enough (another 9% of the time, you'll need a parabola).
You can drop some of the petty stuff (x,y,(180-x-y) triangles), but don't skip the basics.
I'd also suggest skipping rings/fields since most of the algebra important to CS involves structures weaker than groups (semigroups and monoids). Also add graphs/combinatorics to your list.
[1] This fundamental fact is briefly mentioned in a subsection called "differentials", and otherwise ignored by textbooks.
I think you may forget what is taught as "trig" in school. Yes, knowing what sin, cos, and tan are is important. No, wasting weeks on pointless identities and digging into the minutia of lines and triangles well beyond what you will ever need, even in the course of getting a degree in mathematics, is not a sensible use of time. The only reason it is done is "that is how things are done".
When people argue about the math curriculum, they almost always argue the wrong question. The question is not, "Should we cover X?", because in isolation the answer is always yes! Should we cover trig? Yes! Should we cover set theory? Yes! Should we cover graph theory? Yes! etc. etc. The question is, "Given our limited time to allocate to math education, what are the best topics to focus on?", and once you consider the wealth of incredibly valuable topics neglected (elementary economics, elementary discrete math, actual algebra, game theory, computer programming, anything remotely resembling actual mathematical practices rather than memorized formulas stripped of all motivation and history), you'll find that spending umpteen weeks on trig is really shortchanging the students. The opportunity cost of trig is too high.
I freely admit I have no idea what goes into a trig class. I dropped out before trig was taught.
You are probably right: when I teach calculus, there does seem to be an assumption that students know way too much petty nonsense. But some basics are necessary, even if the computer knows how to compute sin and cos. Students must understand angles and straight lines.
As for tradeoffs, I completely agree. I just think the value of basic trig is ridiculously high.
I used a lot of trig in my physics classes - yes, including identities. You may be falling into the trap that because it wasn't useful to you, then it's useful to no one.
And you may be falling into the trap that because it's useful to you, it's useful to lots of people. Again, it's not about "is it valuable", it's about opportunity costs! While you're fiddling with trig you're not learning other more useful things.
Let it be learned when it's actually useful. If you use it in physics, fine, learn it there, when you have context. Not in some abstracted "trig" course.
To be honest, I'm not sure I believe you anyhow. I took a lot of physics, as much as anyone not majoring in it will take, and I did not make heavy use of trig identities, nor did anybody else, nor do I recall a huge number of problems where they would have come in useful, and what problems they might have been useful in were textbook problems anyhow. (In the real world, inclined planes are not all at 30 and 45 degrees.) I think you might just be saying that to score rhetorical points.
But what are the more useful things? That depends heavily on what, exactly, you go into. For many people, most of the math they took is never used. But that's not known at the high school level, so a high school education tries to serve as a foundation for further learning. You're assuming discrete math would be more useful to more students than trig. I doubt this is true.
If I had to learn trig my freshmen and sophomore year of college, when I was taking my introductory physics classes, I never would have kept up with the physics. My course assumed a solid foundation in trig and calculus.
I distinctly remember having to use various properties of triangles to solve many of my introductory mechanics problems.
You can drop some of the petty stuff (x,y,(180-x-y) triangles), but don't skip the basics.
I'd also suggest skipping rings/fields since most of the algebra important to CS involves structures weaker than groups (semigroups and monoids). Also add graphs/combinatorics to your list.
[1] This fundamental fact is briefly mentioned in a subsection called "differentials", and otherwise ignored by textbooks.