This is going to be a difficult question to have a sensible thread about because it involves two totally separate issues - private vs. state and SAHM vs. working mum.
My children are in state schools and I work full time, as does DH. There seem to be a lot of SAHM on MN, but when I looked up the stats on SAHP, only 1 in 8 children come from a household where all the adults in the household don't work. That 1 in 8 is going to include people who are not really SAHPs in the sense we mean it on MN - they are going to include people who are at home because of serious mental health problems and addictions that would make it impossible for them to work regardless of children, and teenage mums who have returned to school/college but are not working. So SAHPs in the true sense of the phrase are a sub-group within that 1 in 8 parents.
So I wonder who the majority are actually getting into this heated debates with. The world is now one of working mothers; that includes my doctor, my children's teachers, my colleagues, my librarian, my dentist, my newsagent, many of the taxi drivers and so on.
These debates are pointless because there is no real fear that working mothers or our daughters are going to have a narrow choice of careers or are going to have to live in a world where men dominate every profession and you can't see a female doctor if you want to. The world has changed.
I think as we don't know these SAHMs on the internet and I meet very view in real life, then the people we are really arguing with are our mothers' generation, who often did stay at home. I may be wrong, but do all the WOHMs who get involved in these threads have a lot of experience of SAHMs of their own generation?
For me personally, as soon as I was old enough, I went out to the library or with my friends after school because I found it incredibly claustrophobic being in a house with someone who had made my life into their job. Your 4 year old may want you at assembly, but does your 10, 12 or 14 year old want you around all the time? I certainly didn't.
I would say, regardless of whether your child is at private or state, as soon as it is possible you should teach them to use public transport or cycle to school and stop picking them up. Teach them independence.
It is all very well being a SAHM if you are like Bonsoir and have a life that is not about your kids, but all this hanging around at school mentioned in the OP is the kind of thing that I find worrying. Even my mother let me make my own way home from primary school.