After teaching proof-writing to my students for several years now, I've seen a lot of variation in how quickly students take to the skill. Some of them have the same experience that it sounds like you and I had, where it "clicks" right away, some of them struggle for a while to figure out what the whole enterprise is even about, and everything in between. Basically everyone gets better at it over time, but for some that can mean spending a decent amount of time feeling kind of lost and frustrated.
And this is a very self-selected group of students: they're all grown-ups who decided to spend their money and spare time learning this stuff in addition to their jobs! For the kind of high school student who just doesn't really think of themselves as a "math person", who isn't already intrinsically motivated by the joy of discovering what makes integrals tick, I think it would be an even harder sell. High school math teachers have a hard job: they have to try to reach students at a pretty wide range of interest and ability levels, and sadly that often leads to a sort of lowest-common-denominator curriculum that doesn't involve a lot of risk-taking.
proof based math requires critical thinking and its a lot harder to scale the teaching of critical thinking. We dont' pay enough for teachers of quality to be able to do this at the public school level. Its also much harder to test for in standardized tests.
You could test it using interactive proof verifiers. This would also make it a lot easier to teach, since proof verifiers can handle even very complex mathematical proof via the repeated application of a mere handful of rules. (The rules are also surprisingly similar to the familiar "plug and chug" workflow of school-level math, only with different underlying objects - lemmas and theorems as opposed to variables and expressions.)
Proof-based math classes came like a revelation to me. When I took Real Analysis, for the first time in over a decade, math was fun. You weren’t just memorizing and reapplying recipes. You were seriously thinking about unique problems and devising solutions. And all the while, you were learning where all these techniques actually came from and how everything connected together.
I don’t understand why we can’t have more proof heavy math in high school. Who cares whether you remember the arctan substitution or whatever in an integral; I’d always just use a solver for that anyway. I’d rather be learning about what an integral is in the first place.