The issue here is that there are a lot of different branches of mathematics, and only one of them is being taught in schools.
Algebra: The concepts behind basic algebra ("Algebra I") are essential for anyone doing any sort of mathematics, so you can't really get rid of this. From what I remember of my Algebra II and Trigonometry classes, nothing there is a prerequisite for any other branches except calculus. It is essential for someone going to college for a scientific degree to know trigonometry, though. It is probably unnecessary to teach everyone trigonometry or "advanced" algebra*.
Geometry: Used as a didactic tool to teach logic, reasoning and proof. "Math without numbers" might attract people who would not otherwise enjoy mathematics. Teaching proof to high school students sounds great conceptually, but from what I have heard attempts are made near-constantly in this direction by math education theoreticians and never catch on. Note that while geometry actually does do the "develop students' logic and reasoning" which everyone says math is good for, algebra generally is not taught this way: rather, it is presented as a series of magical formulas. Certainly there is nothing even close to a "proof-based math course" in high school except geometry.
Calculus: Essential for anyone going into science, unimportant for everyone else. The algebra -> calculus line is the one primarily taught in schools, which is very convenient for future scientists and engineers and annoying for everyone else.
Statistics and probability: A basic understanding of statistics is essential for understanding current events, politics and the news, since statistics are everywhere and one must learn to judge whether they mislead.
Discrete mathematics: Way cooler than calculus, but probably not suitable for high schoolers, as it is not very relevant to people outside mathematics.
"Life economics": I'm not too knowledgeable about this, but from what I have heard, a lot of schools have a "life skills" course. This probably includes basic knowledge about compound interest, buying versus renting, and other economic skills which ultimately stem from mathematics.
Given this overview, I personally came to the conclusion that the best way to teach high schoolers math would be to require a basic algebra course, a geometry course (maybe), and a "useful math" course, which would include basic statistics, basic probability, compound interest, investing, and other ways students will probably actually use math even if they don't go on to become scientists or engineers.
Of course, this does nothing to fix the related problem that many students do not find more theoretical math interesting, limiting the number of people who go into math and science. But as long as most of the math teachers out there don't even like math, we'll just have to live with students not liking it either.