State education and plenty of other luxuries may suit the OP.
I do not want to debate the benefits of Private education I just want to dispute that it is for the super rich.
I know of many families who have paid and are not super rich , but do have tiny mortgages and perhaps enjoy budget holidays.
London independent schools may be different and fees and other costs vary greatly but this is what it would cost to be a day pupil local to me in Kent.
Years R to 2 £7,000 per annum
3-4 £9,000
5-6 £11,000
7 - 13 £18,000 approx
Total at todays prices: approx £187 + uniform + extras.
And there are other private schools in this area with lower fees than these.
In prep extras are not great except for optional music lessons:
Music lessons are approx £40 ph (about £10 more than one would pay privately outside of school in this area).
Non-means tested scholarships are possible anything between 10 to 50%. I certainly also know of people who have negotiated (non-means tested) discounts for their families.
Moving children into private just at 6th form sounds a strange move to me, I don't know anyone who has done that but plenty who have left private for state sixth form.
It is worth noting that the 15 hours free childcare applies to private schools and is up to the end of the term in which the child is 5. So for summer born children the whole of reception year could be subsidised.
Other points worth thinking about related to the budget:
Families can go on holidays at cheaper times as school year is different to state schools.
Uniforms will cost hundreds but be far better quality - last longer.
Long school hours mean that for younger children you might not have to pay the extra cost of after school clubs/ morning clubs if both parents work (for younger children).
The fees may include lunch, snacks, and various activities that parents pay extra for at state schools.
Some schools include Saturdays.