If you have students that cannot understand basic formulas and memorize them, you are not going to teach them abstract concepts. Concrete math is much easier than abstract math for the vast majority of people. I cannot really believe the stories about people who are supposedly talented at math but are not engaged enough by the school curriculum and consequently fail at the tests. School level math is very easy for anyone with some aptitude for math. Anyone who fails it will probably have even greater difficulties with abstract concepts behind it.
That gifted students are not stimulated enough is the real problem. Math classes for gifted children is probably the only solution. Something like Stuyvesant High in New York City but on a wider scale.
As for better math education in Singapore and Taiwan, well, they do have better students than most US public schools. Both genetics and culture play a role here. Asian people have both higher average IQs and higher average conscientiousness. Their teachers are able to teach math at a higher level. US could surely learn from them, but not every school could use their techniques.