[VERY LONG ANSWER, needs patience to read through]
I feel this is a problem many students who are good at maths face. They understand the simple tricks and patterns which are present in the school syllabus and so it is simple for them and after some practise and memorisation they are done. Then they seek out more maths and find out about topics like trigonometry and calculus.
One thing which I feel is a big factor causing the drive and a part of the problem which many people ignore is the 'need to be a genius'. Growing up we all hear stories about Einstein and his legendary E=mc^2 formula which he dreamt up because his brain was just so big. In mathematics we have role models like Euler, Gauss and Ramanujan who seemingly picked out amazing results from thin air. These stories make up our perception about scientists who are introverted geniuses which seem to know everything except how to talk to people and conform to normality.
So students such as myself begin to rebel against the system and find out more advanced maths on their own. However, this is much more difficult than just following the school curriculum. Even if the topic is within your intellectual capacity, having no one to explain it to you is very discouraging and you give up easily. Since you are talking about studying high level mathematics in school you definitely would have experienced finding out about some topic or the other in which you have no idea how to even process the proofs and theorems related to it after a certain basic point. This is very frustrating because you have an image of being good at maths in your mind and you cannot meet it if you don't immediately understand your textbook or whatever you are studying.
This then becomes a trap: you don't study topics which you don't immediately understand or have good intuition for, and so you keep going down and down into deep rabbit holes where you go into one sub-topic after another without pausing at any level to expand your knowledge to related topics and building a firm base before you go on into more specialization.
This is why there is a need to tell students to be firmly confident in their own stupidity. They don't get things immediately and make like 10 mistakes while solving a question, but they can explore and work harder until they are proficient in that topic. Not only their own stupidity, they should know that everyone else is also stupid. Even Euler, Gauss and Ramanujan were stupid, in the sense that each one of them must have struggled with some topic or the other, and they must have felt frustrated and incompetent many times because of that.
The best way I have found to overcome this inferiority complex is to let students make something original of their own. If you know about the process of making an entirely new discovery (new to them, maybe not to the world) without always relying on thought patterns and tricks which are just programmed into you after solving many school and Olympiad level problems (which do not test your mathematical ability accurately), you learn to appreciate many things. You realise how random and arbitrary making progress in a problem is. You can be stuck for days on a single lemma but come up with a one liner that completely resolves it while having a bath. It is also extremely non-linear, which means you can take long detours without coming close to the correct method. However, you also realise that progress in a small or big form will always come if you try hard for long enough (and take enough breaks to reset).
When you realise that every one of those geniuses went through this same random, frustrating but highly satisfying process every time they solved a hard problem, the illusion of being smart only if you are lightning fast in solving and understanding is quickly broken. The only thing you need to make an independent, worthwhile discovery is lots of studying and lots of thinking. That is how maths in the real world is done. You may never get even close to a level like that of Euler, but it is unreasonable to hold such an expectation.
So I think you should encourage your juniors to explore maths: not just topics which are at a higher level, but to truly explore and wonder about things and try to find out more about them. Learning a new topic should be an interest rather than a habit or a drive. It is psychologically very unhealthy and dangerous to have such an unfounded fear of failure and lack of confidence in your abilities. Expanding your horizons above school, competitions and Olympiads helps with that.
As an example, when I proved the formula for the sum of a geometric series in 9th class months before it was taught, I felt much more proud and confident in my abilities than when I solved much harder Olympiad type problems which tested not my independent thinking but my ability to remember and apply obscure formulas and patterns. It also probably helped me much more in developing mathematical thinking than those problems.
As for school, unfortunately, it is a necessary part of life and you have to devote some time to study school maths and other topics according to what marks you want. It is not all bad: if you look close enough, there are many things to be explored in school mathematics.
Finally, when talking about studying other subjects, you should have practicality in mind and know about the consequences of your actions when you avoid studying for Olympiads or school. If you are okay with sacrificing that to fulfill your interest in maths, only then should you do that, otherwise you should search for a compromise.