If we're interested in the mathematical method and its power, a good start might be to, well, go with more exploration-based approaches, where students have to learn things in order to solve a "real"(-ish) problem. Of course, the training required and attention to students would rocket and is likely infeasible, but hey, if Euclid's elements in the original Greek are on the table, this doesn't look too bad !
> more exploration-based approaches
Do you know of any that you'd recommend?
Unfortunately I didn't have this experience until I was an undergraduate. For the prior 12 years of formal schooling, math mostly felt like repetitive drudgery with occasional flashes of excitement when I would learn a new technique.
Edit: I think that was the first class where we learned about proofs and we solved so many that I really get the hang of it.