It's not all drilling for tests in the American system, and this is where some of Chua's observations are lost in translation, because tests are such an important of the system in Britain (and Ireland -- no idea about continental European systems).
In America consistency of effort and homework product from day to day, week to week and month to month is the name of the game thanks to the grade point average system. Your final grade for the semester depends on your participation in class, every piece of homework you have turned in and your tests and quizzes along the way as well as the final semester exam. Ahead of each final semester exam every one of my DCs got notification of the percentage they had to achieve in their final in every subject in order to get an A, B, C or D for their final grade for the semester. A final never counted for more than 40% of the semester grade iirc and in some courses it contributed a much lower percentage to the final grade, so you could not hope to raise your final grade by much in many classes.
And the content of the final was discussed and prepared for in class and at extra study sessions, with the understanding that students would also work at it at home if they needed to. It wasn't a case of making children learn the whole semester's coursework off or guessing right about the exam content in hopes of hitting the jackpot on exam day. The philosophy of teaching is that students will have a solid grasp of the material, and their test taking skills, adrenaline management, or sheer dumb luck are not what the exams are intended to measure.
When parents arrive from a system where the exam is the be all and end all, and they see how the possibility of achievement is built in instead of systematically excluded, it looks like a candy store left unlocked. At least that is what it looked like to me, arriving form the very cutthroat and completely exam-focused Irish system.
US students heading for university do also have to do well in ACT (American College Test) and Scholastic Aptitude Test (SAT) which are aptitude tests as well as tests of material covered. In order to do well, students need to be all rounders as the score is cumulative. So while parents may have a student on their hands who enjoys math or music or English literature and can do well at these subjects without tearing their hair out, keeping them working really hard at subjects they find challenging is a task that ambitious parents have to grapple with.
In such a system, parents who take their eye off the ball for the whole of September and October and allow their children to forget about homework and weekly tests, etc., are going to find their children unable to achieve an A even if they learn the textbook completely by heart the day before the exam.
The three qualities Chua singled out as really important matter greatly in America. Again, I think impulse control is the most important because a consistent and very disciplined approach on the part of your student is really, really important in America, and very few 12, 13, 14, 15, or even 16 year olds have the maturity to manage that on their own (and there are lots of very serious distractions in the middle class communities Chua is familiar with), but I see where she is coming from as far as over-identifying with the American culture being a drawback goes. It pays to see your educational approach and your own educational experience at home as superior. If you do, then at the very least you will have something to compare your child's curriculum with, and this matters in a world where your child's competition for that place in Yale is a child from Eton or some education factory in Shanghai.
I don't think it matters for the sake of Chua's argument that Nigerians, Lebanese, Mormons, etc., are not the majority in superselective schools (and superselective schools are not a factor in America anyway). It's the high achieving students as a percentage of their own group that Chua is deriving her figures from and her arguments too. The percentage of 'American' students achieving great results is far lower as their group is much larger, obviously.
Speaking from my own experience, of 23 valedictorians in DD2's graduating class (in her HS, this meant students who got all As in all honours subjects for four years running/seven of eight semesters - see description above of the consistent effort across the board this required) two were the children of Russian immigrants (of three identifiable children of Russian parents), one Irish (of two known Irish parents), one English (one known English family), one Indian (four Indian families). Harder to tell who may have been Jewish or Mormon.
In the case of those HS students who achieved National Merit Finalist status based on their success in the PSAT/NMSQT exam (these students came in the top half of one percent nationwide in that particular exam) 17 from this particular graduating class of 850 representation of children of immigrants was even more pronounced as a percentage of the whole (and there was a lot of crossover between valedictorians and National Merit Finalists, as you might expect.)