I don’t have time to read the whole thread so my post may be out of line, but here it goes.
Seems there is a lot of misunderstanding around these competitions, mostly among people who know nothing about them. These are not “obscure” but the most prestigious events in the world on HS level if you are interested in STEM.
In our immigrant group of friends, 4 participated in math and physics Olympiads in the early 80ies (in a different country, not US and not in Asia). All got full scholarships to the most prestigious graduate programs here (Caltech and the likes), ended up holding multiple patterns and are very successful in the industry and academia. Most importantly, they are terrific thinkers and problem-solver on multiple levels.
Firstly, the kids who go to the top in these competitions have a natural gift and interest in the subject. They often start reading, experimenting, doing problems for fun at a very young age without any prompting from their parents. Often, their talent is nurtured by a great teacher or mentor, and they are selected for magnet schools or group such as math/physics circles and summer camps. It is impossible for a parent to turn their kid into an academic prodigy, any more than turning an average athletic kid into a junior Olympian.
Second, the material covered in these competitions is not some obscure topic that they will never come across later in life. On the contrary, these competitions go more in depth of the bare bone curriculum offered in the US schools where math kids are rushed to learn calculus in 9th or 10th grade ending up with superficial knowledge. I am talking about public schools as this is where my kids went – a supposedly top public HS. In the country where I and the above-mentioned 4 olympians grew up, they teach calculus and linear algebra in college, not HS. The physics Olympiads for instance, force you to understand mechanics at a much deeper level than most college classes, and certainly better than the 1 semester that you spend in AP Physics C. You learn best by doing problems.
Thirdly, even if your kid is not a prodigy and will not go to the top, the knowledge and skills they acquire will serve them for life: grit, perseverance, ability to focus and tackle very difficult problems, mental connection between concepts, ability to apply an abstract concept to solve a real problem, lateral thinking, out-of-the-box thinking, ability to explain difficult problems when you discuss them with peers, team building when you solve a problem together with your circle buddies.
My son did the competitions, and gain all of the above. He did not put more time into this than somebody playing piano for instance. He is extremely social, plays a sport and goes skiing regularly.