"impression that more advanced statistics is technical elaboration that doesn't offer major additional insights"

Why did you have this impression?

Sorry for the off-topic, but I see this a lot in LessWrong (as a casual reader). People seem to focus on textual, deep-sounding, wow-inducing expositions, but often dislike the technicalities, getting hands dirty with actually understanding calculations, equations, formulas, details of algorithms etc (calculations that don't tickle those wow-receptors that we all have). As if these were merely some minor additions over the really important big picture view. As I see it this movement seems to try to build up a new backbone of knowledge from scratch. But doing this they repeat the mistakes of the past philosophers. For example going for the "deep", outlook-transforming texts that often give a delusional feeling of "oh now I understand the whole world". It's easy to have wow-moments without actually having understood something new.

So yes, PCA is useful and most statistics and maths and computer science is useful for understanding stuff. But then you swing to the other extreme and say "ideas from advanced statistics are essential for reasoning about the world, even on a day-to-day level". Tell me how exactly you're planning to use PCA day-to-day? I think you may mean you want to use some "insight" that you gained from it. But I'm not sure what that would be. It seems to be a cartoonish distortion that makes it fit into an ideology.

Anyway, mainstream machine learning is very useful. And it's usually much more intricate and complicated than to be able to produce a deep everyday insight out of it. I think the sooner you lose the need for everything to resonate deeply or have a concise insightful summary, the better.