I don't actually think it's that extravagant - those feeding 5/6 where 3/4 of them are DC's, and spending less - age of DC's makes a huge difference.
It is easy to spend less on food if you are feeding 2 preschoolers and 2 infant age DC's.
Once you are feeding two adults, two teenagers that eat MORE than most adults plus two older primary DC's, I defy you to feed a balanced diet on less than £160 a week if you are including cleaning products and toiletries in that.
A teenager will always eat about 4 times as much food as a fully grown adult toddler, and it DOES push up the food bill.
And that works out to £8320 a year. So there's only £1680 a year 'wiggle room' in their shopping budget - only about £32 a week.
Maybe bulk buying washing powder when it's on offer, buying cheaper meat could account for that (we never eat lamb or steak, for example, to keep food bills down), but I don't think a £30 a week saving is going to help them feel any 'less skint' tbh.
It's the private school that is the highest unnecessary cost.
Food is an essential cost, and I can see that if the OP's DH is working long hours for that £250k, he's not going to want to give up the occasional steak dinner or lamb chops dinner...
Until your DC's have finished school, you are going to feel the effects of your DH's choice to send them to private school. But as it is his hard work that is paying the school fees, I can't see where the problem is.
Your DC's will have had the advantage of going to private school.
It's no different to me not paying for Sky TV (don't have it) or my Internet (my uncle pays it, long story) so that I can pay for an after school club for each of my DC's.
I can see the advantages of going without now so that my DC's get the long term benefit of having been involved in an after-school activity.
Ok, so I don't have certain things - but it is to benefit my DC's.
It's priorities.
I think you are feeling frustrated because you and your DH have different priorities where private school is concerned, and it isn't as essential an expense from the household budget for you as it is to him.
And it DOES grate more when spending priorities differ in a couple and it means one person in the couple going without what they perceive as the benefits of that income.
Maybe you need to discuss again the Private school issue with your DH?