You are asking all the right questions @M0msthew0rd! We had the same questions.
I don't have direct knowledge of the Perse, except from other parents and the children but by all accounts it is a great school. It does have the reputation of being "high pressure" academically competitive compared to the other private schools, but it boils down to a couple of questions.
Are one's children competitive? Do you have the time and willpower to spend time helping the children with their homework? You only get the best out of private school if the parents get involved, you can't expect the school to do it all.
Our daughter needed help academically and while bright she was let down before and kings identified this within a week and are giving her all kinds of help and she is improving at a impressive clip. Our son, who is competitive takes pride in his homework and gets good results. He was thrown in at the deep end but swims well. He wasn't like this before.
The advantage of the Perse is that it does go all the up up to A level, whereas we will have to find a new school for both after 12+.
We had a similar experience, like you, with the tour of Kings and found the staff to be welcoming and warm. We did have the option of St Faith's but geographically Kings was easier for us in terms of pick up/drop off.
For us, Kings has worked wonders and I can only speak favorably about it.