Name changed for this. I went to the Perse Upper in the 2010s, but am very much still in the loop about it for various reasons. I found the staff really brilliant - hugely devoted to their subjects, with many going above and beyond for students. Also, the facilities are really lovely. The school itself felt humanising and empowering, without too much micromanaging crap about minor uniform violations and similar.
However, the academic pressure is insane, and academic ability and work ethic alone are not enough. Socially and culturally, the transition from a very ordinary state school was extremely difficult. As nice as it was to no longer be in an environment where reading a book was "posh" and doing well was embarrassing, all my peers had attended competitive "hot-house"-style prep schools with one another, so I was totally unprepared. There was definitely some sneering and ostracisation from some of the "prepped" kids towards some of the state-schooled kids, especially those of us from non-academic families. The school came down swiftly on bullying but ultimately, you can't force teenagers to be accepting.
DC who went to state primary but had Oxbridge/privately-educated parents would probably be fine, but otherwise it can be tough. There is an extreme focus on Oxbridge and medicine and a strong expectation to achieve a clean sweep of A*s or equivalent. I have definitely noticed that it's easy to lose a sense of perspective when in this kind of environment - for example, bottom set in the Perse is realistically still higher than top set in almost any other school.
I view the "alpha parent" phenomenon as the product of the Cambridge middle-class, not the Perse itself, but the Perse is a major hub for "alpha parents", as those upthread have called them. I'm very much not trying to deter you from applying for the Perse - it's a really nice school with a progressive feel to it. I would caution that you might want to put DC in a feeder prep from around year 4 or earlier - then again, as you were privately educated yourself, you will probably be somewhat able to prepare DC culturally and academically, but it's worth considering. I have also observed a lot of parents place a huge amount of pressure on children to pass entrance exams - if DC needs intensive private tutoring to pass an entrance exam, it only gets harder once they get into the school. In my (very humbly offered) opinion, it's far better to take a "let's see" approach and apply for a few schools with various selection criteria rather than honing in on getting DC into the Perse by any means. As I hope is clear, I'm just hoping to give some helpful advice, as I have a fairly positive view of the Perse.