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.