As I said, in Sutton's first round, white children still make up a large group, maybe around 30 to 40 percent, so they do apply. It is just that you rarely see them make it through to the second round. For super-selective schools like Henrietta Barnett, it is different because the proportion of minority students is already so high and the exams are so competitive.
I do not think tiger parenting is absent among white people either. Look at London private prep schools and selective indies: there is no less tiger parenting there, even while those schools are very diverse and white students are the majority. True, working class white families may not place the same heavy emphasis on academics, but I also do not think tiger parenting is the only way to get into selective schools. I agree that over-preparation is toxic and does no one any good. However, framing 11-plus prep as a binary choice, where choosing not to prepare is a brave defense of a happy childhood and everyone who passes must have sacrificed theirs, is far too simplistic.
If your child is grammar school material, give them proper preparation and let them have a go. Do not fear your child being a cultural minority in a grammar school, just as minority families previously embraced going to predominantly white schools. That is real bravery. And this would help the grammars become more diverse!