There is no single right answer to this. As adults, though, we need to weigh up what would create the greatest 'sum of happiness' for the whole family.
So if, for example, a great part of your personal happiness and self-worth is linked to your job / career, then the loss of that would not compensate for the marginal benefit that the children might receive through you being at home more.
Or if you need to work 12 hours a day in order to provide a roof, food, warmth and clothing, then again the benefit of that far outweighs any disbenefit to the children of you being out of the home.
['You' by the way, is purposely gender-neutral - it could equally apply to male or female parents]
On the other hand, the marginal benefit to an already financially-secure family of being able to provide private schooling for several years vs spending more time with the children when very young is a much more finely balanced 'sum of happiness' equation, in which the balance is tipped by other factors such as current main caregiver, quality of alternative schools etc.
Every family will have a different 'sum of happiness' equation.
Just as an idea, though - you mention moving to a different, perhaps 11+, area once the children are secondary school age, in order to get round the 'only able to afford private for primary' issue. Could you not plan to make a move much earlier, to an area with great state primaries and good secondaries (whether selective or not) and therefore take yourself out of the binary 'work excessive hours / need to pay for schooling' vs 'work fewer hours but be limited to less good state options' dilemma?