Math education in school up to very, very late in undergrad is extremely disconnected from what real mathematicians do. There is the very reasonable focus on preparing students for what is a far more likely and pragmatic usage in applied STEM disciplines, but surely there is something regrettable about students coming away with the impression that being good at math means being good at reproducing a known procedure. Well, not that that's seen as a problem. I guess it really doesn't matter..

Anyhow, I'm not the firebrand that Paul Lockhart of A Mathematician's Lament is/was. I think that an achievable concession would be a "Mathematical Thinking" class as an elective for high schoolers. This class would be close to Lockhart's proposed model where there might be an initial problem that the class collectively tries to work out, and then break outs into groups of students that each work at chosen subfields and a combination of questions they come up with themselves or are suggested by the instructor relating to properties of basic geometry or integers and whatnot. Utilizing "naive" approaches might be fine here as the emphasis is on reasoning and understanding. You need very little "theory" to reach spaces with a fair amount of complexity and overlapping interactions.

Some amount of history could be taught to to introduce the various problems that mathematicians faced at different times in history and their "inventions" that became major contributions to mathematics--Euclid's geometry, polynomials, calculus, etc.

Additionally it could be helpful to introduce set notation and more rigorous definitions. It's a different (and honestly, it's not math if it's not tightly defined, in a very literal sense) way to think about mathematics, but those basics are not any intrinsically harder.

Maybe they could use computers and online editors like ShaderToy and Desmos to create visualizations as well. That sort of thing has all sorts of interesting spaces of exploration.

Ultimately the problem is that applied math is much more relevant in cases by a factor of probably 10K to 1 so there really is just no point in making a larger reform beyond that. Also there is the added difficulty of needing a instructor who's pretty good at teaching something like this, which is a lot more challenging than the conventional setup. And I wonder if that many students would really be interested in signing up for a class like that..