OP - the bit that sticks with me, you and your brother decided that no child of your family should go to a modern as you had a shit time, and your sister then later 'agreed' with this - but clearly wasn't one of those who felt so strongly and passionately about it.
You tied your financial life to your parents and your brother, to your extended family and do seem to see family money and resources as shared, putting the family business first. But you, your DB and parents seem to miss that your DSis thinks differently, she didn't take from the family business, she got into a grammer which you didn't, but she did it from her own abilities, then didn't go to uni, but worked hard, built a career and financial position for herself that wasn't due in any part to your parents.
She didn't see her money as a shared resource as it hadn't been funded in a shared way.
As you would feel all family members have a right to use your assets and financial position in order to do what is for 'the greater good', you obviously assumed other family members money and assets were the same - but as you can see, you and your parents are the unusual ones, the bulk of population don't see money as a shared asset in the extended family, and your sister holds a perfectly valid position that her money, that she earned herself, not in the family business, as hers, not yours.
She also obviously didn't share your view that "moderns" are exactly the same as they were 20-30 years ago and are automatically a disaster for a young person to be sent to one.
Indeed, your DD1 when faced with the choice of moving didn't want to, and still has gone on to prove your sister right and you and your DB wrong by doing so well. Your DD1 didn't need private education.
If you think you can build a better life for your DCs to move to near your DB, then you should go for it, but as it does seem to cause you such angst, have you considered avoiding Trafford with the 11+? you might be better with a good comp, there are lots in Cheshire which can give you a garentee if you move into the area in time, you'll get your DS in. Far less stressful for you, it does seem that you have massive issues with the fact you 'failed' at 11, your DD1 'failed' (even though it looks like she's been fine!) and your dsis 'failed' by not seeing it as the disaster you and your DB thought it was.
You won't heal things with your Dsis until you accept that actually, she had a right to not hand over all her savings, her only security for something that she didn't see as important, only you did- that she's entitled to consider the money she earned without help from any of you as hers, not a communal resource.