I have some of the same attitudes to some mathematics. But there are two strong forces against me.
First, the curriculum is crammed with gumpf. There is a small set of mathematical knowledge that I think is important for citizens of a developed democracy to know, stuff around finance and statistical reasoning mostly. But I could probably cover this in one semester in Year 10. And there is a small amount of foundational number knowledge that makes it possible to teach much of the rest of mathematics - times tables, an understanding of place value. Again if this were done carefully it could be done in about a semester - I'd prefer if it were done in primary school. But I have to spend an awful lot of my time teaching other stuff that is not in any sense necessary or useful, coordinate geometry, trigonometry, calculus, volumes of complex shapes, multi-variable algebra etc etc etc. Any one of these would be fun to go into in some depth but the necessity of covering them all means that none of them are covered properly and the connections between different areas of mathematics are totally obscured.
Secondly, my students all come to me with a history of mathematics classes. Mostly, this history teaches them that there is a right answer and they are too stupid to find it. They wait to be told, they attempt to memorise formulae and they lack curiosity about how things work. I make attempts to reverse this but when the rubber hits the road and I need to cover content quickly, I reinforce it despite my best intentions.
If someone wants to found a charter school where I can use Godel Escher Bach as my only maths textbook just tell me where - I'll catch the next plane.
"Everything should be made as simple as possible, but not simpler." -- Albert Einstein