There was a system like that. Roll up, roll up, I'll treat you to a story:
It all started in 1957, with the Sputnik.
The US was entirely taken by surprise. The belief that the Russians were way ahead instigated an internal crisis, which, in turn, led the US to re-evaluate its national maths curriculum. Thus was born a think tank called the School Mathematics Study Group (SMSG). They developed a radical reform in mathematics education known as 'New Math'.
This was rolled out nationally, to great criticism. The teachers were ill prepared and the parents felt clueless. Look it up, there were 'Peanuts' strips from the period mocking it.
Now, New Maths focused initially on the early years, but then, in the mid '60s came a second round, and one specific initiative was the Secondary School Mathematics Curriculum Improvement Study (SSMCIS), and the guy who heads it happened to be a professor at Columbia University Teachers College. This last tiny fact probably means exactly zero to you, but it is central to why I even know all this stuff.
Anyways...The program's signature goal was to create a unified treatment of mathematics, so that instead of studying the normal curriculum you'd basically study maths the way you're taught it at university: set theory, group theory, axioms proofs and logic, all the way up to calculus. The programme was intended for grades 7 through 12, and was rolled out initially in the NY area and then later in select schools in other affluent cities in the states. It only ever targeted the top 15-20 students in the a class body. That was for sure the right call - this ties to your question, so more on that in a minute.
Eventually two things happened: one, the programme ran out of funding. Two, by the mid-seventies there was a massive backlash against New Maths and the US decided maybe it's okay to just leave it, since the Russians didn't end up winning the space race after all.
I would have known diddly squat about this whole affair were it not for a curious corollary. In 1953 one very specific individual happened to be on a mission in New York. He was a former Russian Jew who studied maths in Canada, served in WWII for the US military and eventually made it to Israel. He was an educator and had somehow caught on to what was happening in Teachers College, and upon his return to Israel, he started devising maths curriculums and translating the original SSMCIS textbooks. This was now dubbed 'The Columbia Programme'.
Fast forward almost 40 years later. In a way I've never managed to uncover, that programme survived, and was still being taught in one of Israel's gifted programmes. I entered my first maths lesson at seventh grade never realizing just how much this would end up influencing the person I'd become. Our textbooks were literally photocopies of the typewritten texts. The teachers has added to it bits of the regular curriculum plus more practice exercises, which the original textbooks lacked, but they left most of it as is.
In the first three years, no one in the class was allowed to drop out and take 'regular' maths. For many, even in a cohort that was already pre-screened for academic achievement, this was a struggle. For sure, once highschool rolled along, anyone who hated it could switch back to the regular national curriculum.
Of the people who stayed, nearly everyone went on to study Maths, Physics or Computer Science to graduate level. This tended to happen in the years above and below as well. Over the years, though, the programme got smaller and smaller. I'm not sure it still exists.
To your question:
You absolutely CAN get highschool students to leave secondary school with advanced-undergraduate level of mathematical maturity. (And, BTW, the Russians are still ahead there...) But you can only do it for a small minority. Not because of elitism, but because most people aren't a good fit for this path.
At the time I intended to write up all of this into a nonfiction essay. But other things took greater priority, and I just left it there. In a way, it's been nice telling this story here.