Tiger parents come from all over, Europeans as well. Inevitable really,
when the City collects over-achievers from across the world who are determined that their children will follow similar paths.
How much you push depends in part on how much value you attach to an Oxbridge/Ivy place. My sense is that admissions people are so used to seeing "perfect" applicants from top schools that those with the odd B are pretty automatically ruled out. Often their loss. Relatively few of DCs sciency friends ended up at Oxbridge, with most making their way to London, despite the fact that over 50% of the year group would have been Oxbridge/Ivy. Were they less bright? I doubt it. Clearly there were some very bright/special kids who got Oxbridge, but there were also several slightly dull sloggers. And some of those who went elsewhere, quirky brilliant kids who used their teenage years to explore and develop their interests rather than do too much revision, have done very well indeed.
And there are differences. In DS' subject, without exception, those that got Cambridge would have interviewed very well indeed. Their parents were senior City folk, and they were destined to be senior City folk and would have had the confidence that went with it. Most other Universities do not interview for that subject, and given how well so many of DS' peers did, I doubt that his cohort was any less able. I am particularly proud that DS' UG tutor had no idea which type of school DS went to till she saw his final results just before his graduation. (She was American and went to a similar school in the US so thought she recognised the type.) I suspect that, just as it is difficult for academic schools to sift the very prepared from the natually able, it is difficult for Universities to work out which of their public school products sitting in front of them is the real deal. Easier to simply focus on those with the straight As, even if in some cases those As will be the result of a pretty intense tutoring schedule on top of a good education. (Though most have done fine, and as educated young people with a great work ethic, will be very employable, we did see a few car crashes along the way.)
London suited DS. His schooling had left him with an interest and engagement in education which meant that he was happy to get involved in a range of stuff beyond his course, and given the number of people who pass through London, he had exposure to some fantastic stuff. May Balls/punting were not really his thing. DD too felt she did not need to try for Oxbridge, though she would have been a credible applicant. She has been surprised to find herself missing some of the scarey ambition and focus that she thought she wanted to get away from, but I suspect that is a first year problem and that as she finds her tribe she will settle on a happy balance of work and play.
My preference too is to see it as a long term game. Focus on encouraging your child to enjoy education, but also encourage them to develop important skills like self-discipline, their own ambitions, and resilience. Being able to try things and fail, then try again is as important a life skill as straight As. Whilst the ability to be happy and to be kind and supportive to others is crucial - who knows what life will throw at our DC. Good enough in the context of this type of school is very good indeed. (I read today, though I am not too sure I believe it, that 51% of university applicants have 3 Grade Ds or worse.)
However Oxbridge/Ivy is very important to some people, including those for whom London does not appeal, and for some careers. If so, I would be tempted to help ensure that you don't give an admissions tutor any excuse to weed your DC out before interview. The same will not apply to those from more academically diverse schools. Its just if 70% of A levels and 90% of GCSEs in the school are at A*, anything less than that looks poor.