Sorry for the delay. I saw this post and thought about replying later, but I forgot. The last sentence is very good advice. But I prefer to use "and" instead of "or".
> "do thousands of logic and math problems and gain an instinct for it" AND "have any extremely high IQ".
Since not everyone can be Tao[1], you and me must try with the other part of the trick. (And probably Tao also solved a few thousand of problems.)
It's like cycling. Not everyone can be be the word champion, but the rest of us can use a bike to work, or go to work, or just to enjoy.
I think it's important a mix of theory and practice. You must read some books to learn old stuff and not reinvent the wheel. But too many books is also bad.
About problems, it's very useful to solve a lot of them. Many tricks can be reused, so sometimes it can look like a brilliant idea, but it's actually a variant of a trick the person used to solve a problem a few years ago, and in same case just a standard trick in the area. Also, sometimes a whole problem can be reused as a trick in a future problem.
The difficult part is choosing the right difficulty of the problems. If they are too difficult, you will not solve them and get unhappy, and even reading the solution is not useful. If they are too easy, you can solve a million of them and not learn new tricks or insight.
(If you read the solution of a problem. Try to rewrite it on your own later.)
I like to use Sudoku to explain what is math to non mathematician. You must think a lot, and distinguish what you can prove it's true and what you wish.
(Protip: Only write a number if you are 100% sure must be there.)
(Protip 2: In advanced problems in Sudoku and math, sometimes you are not sure about a number but you only have 2 options. Make a photocopy, and solve both versions.)
Anyway, the nice thing about Sudoku is that there are some books that have the problems arranged by difficulty. If you try to begin from the end of the book, you will not solve too many boards. Starting from the beginning is sometimes boring and repetitive. The trick is to detect the exact page of the book where the problems are very difficult for you but solvable.
[1] https://en.wikipedia.org/wiki/Terence_Tao