I would focus more on the problem solving side of maths, such as maths competitions

I am still a high school student so sorry I say something stupid. Please educate me!

I am very much against this. I agree on the "teaching them how to solve problems", especially in respect to totally unseen problems or tricky things but I do not agree that math competitions are anywhere relevant.

I am grinding HMMT/SMT/Putnam papers a bit these days since I have to attend a scholarship exam and then the thing is - there is a lot of memorization that is involved no matter how much you claim that you don't need to memorise anything. E.g, Good luck figuring out trinomial revision or any of the discrete difference binomials if you don't know them from earlier.

In theory it sounds great that you can derive whatever you need in the time (AND this is what is needed to be taught) but when you are writing a paper in 3 hours and you don't know what the Weierstrass' substitution is or how does SFFT or any of the Aurifeuillean factorizations work, you are at a significant disadvantage. You will be foolishly spending a few minutes deriving Viete formulas to just use that one simple result.

Also, not everyone wants to do competition math. It is fine if someone who wants to pursue algebraic geometry or differential topology does not have any interest in figuring out random counting nonsense or proving existence of nonsensical representations.

What needs to be changed is that students should be exposed to more specialized math and encouraged to pursue it while simultaneously given more challenging non-routing problems.

tl;dr - Math Competitions should absolutely be a niche instead of a mainstream forced activity. Maths is never about rote memorization but without knowing prior results of the centuries of advancements in hopes of "figuring things by myself", you are only hindering yourself.