<semi-sarcastic-rant />This is what [one of many things] is wrong with the educational system! Sure it's amazing that public schools have collectively taught people basic algebra (in theory), but we should set the new bar to Calculus and just keep pushing it up every so often. (I think Bayesian stats are more difficult than straight-up calculus though I'd like to see a public familiarity with both...)
Calculus has been around for a pretty long time, there's plenty of good material out there to learn it and people who understand it to teach it, there's no real reason it shouldn't be required curriculum these days.
I would NEVER put calculus into required curriculum. Calculus is far less important than some simple number theory, logic, and the ability to think about abstract problems creatively. The tragedy is that we push towards calculus (which is really just advanced arithmetic until you get to, say college, real analysis) at the expense of these other skills.
While agreeing with the sister-child I'll also say that yes, other math and thinking skills are also important, but there's no reason we can't have those and calculus too. Throw out some of the more useless things like PE, get rid of certain requirements like art, maybe take away a year or two of the "read books most of the class doesn't care about and write essays hurriedly read over by the prof to later be never read again" English classes.
Even just removing one required semester (half a year) of PE and one required semester of English (which shouldn't be very controversial) frees up two semesters, which for a high school level accommodates Calculus just fine. I'd love to see mathematical analysis courses taught at the high school level too (especially since many grown adults are under the impression math in general is like the algebra they did in high school) but that might be getting ahead of ourselves.
I also don't see why you can't teach number theory et al. throughout all the math courses. I never had a formal trigonometry course because it was taught along the way.
While I agree with your overall idea of removing extra classes from high school, you do realize that most graduates need at least 2 more semesters of English? Plus the failure rate of College Algebra is pretty staggering.
IMHO you don't understand Calculus good enough yourself. It's one of the most powerful tools humankind has come with. A couple of simple ideas aplicable practically everywhere.
Calculus has been around for a pretty long time, there's plenty of good material out there to learn it and people who understand it to teach it, there's no real reason it shouldn't be required curriculum these days.