Would they?  There are plenty of rich and powerful who send their children to state schools, yet the reason many of the schools their children attend are good has got very little to do with them, and a lot more to do with the fact that they are often amalgamations of previously good schools with previously reasonable schools; good teaching and admin. staff; location; raw 'material'; access to outside influences, such as work experience opportunities, and places to visit; and other factors.  Rather than blaming 'teaching standards' many ofd which are far higher than when you and I were at school, blame the politicians for their constant tinkering with curricula, educational policy, ideas of best practice, etc.

Sigh, Andy, you are showing that you still haven't grasped it. It doesn't matter that some (a few) of the rich and powerful send their children to state schools. The majority don't. These people are not idiots.  They choose private education because it buys them something - some of what it buys them is a better academic standard, some of what it buys them is status, an education where they are taught that they are the elite, the ruling class, where they are taught all those little nuances that allow them to be prosper.  It's a divisive system where the common people's education is looked down upon even if can be just as good, where the old-boy network flourishes, and the same families, same crowd with the same outlook keep and grow their wealth and their power and pass it on from generation to generation.  You have to undo such a system before you have a chance of fixing the problems it creates.