I'm not sure this is wise, for two reasons. First, linear algebra and calculus are remarkably useless for people who aren't headed for STEM occupations. They are vital for some engineers and scientists, who use them all the time, but don't get much use outside these disciplines. I work in a pretty technical domain -- I'm a software developer -- but I've used calculus exactly once outside of schoolwork and linear algebra never at all.
Second, universities usually insist of teaching these two topics anyway. In STEM programs, they are typically covered in mandatory courses in the first year. Students whose high schools did a good job get to coast for half a year, while the less fortunate catch up, which is just wasted time.
If we eliminated linear algebra and calculus from high school curricula, leaving them to those who are headed for professions where they are actually useful, we could replace these courses with some other forms of mathematics. There are plenty of choices: abstract algebra, topology, geometry, combinatorics, and the list goes on.
Of these, formal logic and statistics are particularly promising. Formal logic attempts to rigorously determine the truth value of claims, while statistics handles approximations and drawing inferences from observed behavior. As such, they are particularly useful for sorting through the mass of questionable claims we all face in an information society, to find the truth among the wishful thinking, half-truths, and outright lies. This makes them valuable aids to good citizenship, which is something we should be aiming everyone towards. And as actual domains of mathematics, they can be made as rigorous and challenging as the less useful courses they would be replacing.
