Commenting as a parent with a child who recently went to primary here, but has since been moved. For my daughter's year group, there wasn't the kind of parent drama you are concerned about. Albeit there weren't any celebrities in her year group either. I found most of the parents very nice and I became close friends with a few of them. Some could be a bit stand-offish, but friendly enough.
A majority of parents are what you'd see anywhere else in North London (many parents raised working or middle class), and then there are a good mix of hippy-dippy with world-views not based in reality (think Gwyneth Paltrow "Goop" type stuff). The latter parenting group in particular have "free-range" parenting styles, which didn't align with my values in how I think a child should be raised. So this does mean that there can poor behaviour from the kids that goes unaddressed at home, and then also happens in school. And it also means a portion of the kids are exposed to things that are not age-appropriate, which then makes it way through the entire class eventually. As an example, for their first PGA trip (Yr 4 I think), the girls were all talking about what boys they were going to kiss. Thankfully my daughter told me about this before the trip and I brought it to the attention of the Head, who then had to arrange for the teacher to do a special talk about the topic to ensure nothing happened. You'll see this higher level of being too mature for their age all the way up to secondary school. If you look at the older girls, for example, you'll notice they're wearing lots of make-up, belly shirts, etc.
I really believe in the ethos of the school, and they do an amazing job exposing the kids to all sorts of interesting topics and disciplines. I really love the head of the lower school, too. But unfortunately, I don't think they do a good job with pastoral care. My child is super sensitive with anxiety issues and was bullied from a very young age in such subtle ways that I didn't even realize it was happening for several years. The school did try to help her, but there were some repeat issues with the same kids over and over again, and nothing was done about those kids (they made it more an issue of helping my daughter deal with conflict rather than stamping out the bad behaviour of other kids). So for her mental health, I had to pull her out. It was only after I pulled her out and put her into a very small, structured and supportive school where there really is a zero-bullying policy, that I also realized how much they did not support her academically either. Her current school communicates with us frequently about how she's doing AND they've put academic interventions in place to help her with the things she's really struggling with. Whereas at KAS we rarely had any idea how she was doing (I do want to call out that there was one year where we did know a lot more about her progress, but that's because the teacher was so amazing and kept us in the loop). I realize they do this because they are child-led and generally let each child progress at their own pace. But in practice, this translated to letting her struggle too much and fall very behind in certain disciplines.
I don't want this to come across as bad-mouthing the school. There is much to love about it, including the facilities! But I think this school is not for everyone. The kids I saw thriving there were the outgoing, strong-minded, independent, and self-sufficient kids. Or at least the easy-going ones who are not bothered by much. It's not a school for kids who are sensitive and need a lot of emotional support, and it's not for the ones that struggle academically, as they're likely to fall to the bottom without a lot of attention or support.
On this latter point, if academics do matter to you, I recommend looking at their GCSE results as a comparison. There are plenty of state schools (typically the ones who are rated outstanding) who produce just as good results, and it's free!
In our year group, there was a lot of turn-over for many of the reasons I've mentioned above. So I'm not the only one who had these issues.
OP must have made a decision by now, as I'm posting 1.5 years later, but thought it might be helpful to other parents considering KAS.