Things exist on a spectrum and I don't think we're really on the same page on what it means to be immersed - it's about how your subconscious mind decides to prioritize things, not about any kind of curriculum. I spent a lot of time in my life cleaning and organizing stuff, but I'm not good at it and not getting better at it either. I'm pretty sure this isn't because some lack of natural ability prevents me from being good - for whatever reason, this is just something that I don't find interesting. This is how most people approach math - as a chore to be done and forgotten. Even regular good-at-math people (which most people at elite schools would qualify) that aren't remotely close to competing at the likes of IMO and Putnam are qualitatively different in this regard compared to average students because they are broadly speaking more immersed in their studies. For most people, it's not merely that they don't think about math outside of the classroom - their mind is also too busy thinking about anything but math while doing math.
The flaw with this model is that it doesn't really explain the world, because if you compared most kids who do well on math contests to the average high school honors student, you'd have to argue that their performance in almost every subject besides Band and PhysEd is the result of mental prioritization. Also, observing that an underperformer just didn't mentally prioritize it the same is basically mind-reading; it isn't a refutable observation. We can see the weakness of the argument by using it to explain the effect of age on performance, without finding direct evidence to refute it. But the actual cause is that kids get smarter as they get older. It also fails to explain why kids rank so differently at contests like 24, versus later contests that test a mostly different sort of ability. That kids who do well on math contests like thinking about math can't be taken as a factor in more than a banal sense.
And imagine trying to make the same argument for basketball. Kids with low field goal percentages just don't want it enough?
