I don't think it's problematic to view families as having the most important role - they create the foundations from which you can build something good, and you can't build anything good without strong foundations, you can only create a facade. And basic values have little to do with class or culture. I would reject any idea of a class which did not care for its offspring or wish to pass on its skills and an interest in the world around them. And passing on a sense of insecurity and rigid closed-mindedness and rejection of alternative views is a failure of a culture or class, too - protectionism at its worst. Cultures which behave like this reject the idea of education by a state which is not part of their culture, anyway. Basically, if you can't even accept that people of different cultures need to become more open minded and accepting of other beliefs, then you really can't fit a diverse group of people together under one roof. If a society cannot agree even a very basic set of values shared by everyone, it cannot create a universal education system. That is precisely why the whole concept of a state education is contentious.
And as long as the reaction of other groups to a perceived "elite" is to reject all the ideas of the elite in response to the perceived rejection of their own views and activities, you are going to have a mess. There is nothing that is not worthwhile about classical music, or ballet, or museums, or the artwork and architecture of churches and cathedrals and stately homes (created in any event by the working classes, yet peculiarly appreciated more by the elite - fancy rejecting your own handiwork rather than admiring it for the superior skills that the working classes had in the past, even if they were unfairly used or collected for the benefit of a small minority which precluded its very creators from its enjoyment...), theatres and art galleries. Even when these things are made free for people, or affordable, there is a very peculiar rejection of them by some as boring, elitist and not of interest to the majority. Why do some classes and cultures view them as not being relevant to their lives, or some attempt at brainwashing and squashing everything else out if they are introduced to them?
Yet going the other way, there do seem to have been attempts to accept and take part in the activities and interests of other classes and cultures - pop music, by its very name, is more popular than classical music (and pop concerts are not exactly cheap if you want to go to one - you could go to see classical music being played locally for far less money); football is more popular than rugby or cricket or lacrosse or fencing (but is it any cheaper to attend a top football match????); musicals are more popular than opera; cinemas are more popular than museums. It's a bit rich when people complain about something being elitist and then say they don't want it anyway if it is made accessible to them.
It seems to me that some cultures and classes which feel failed by the education system only want to be educated in the things they already have and know a bit about, which doesn't seem like much of an education to me. It is those who are not part of their class or culture who need educating in it - but you cannot provide any kind of depth or quality of understanding in your education system if you do try to cover a little bit about every possible culture in the world, or every culture represented by someone in this country. So you have to pick and choose to a certain extent - and that means that more time will inevitably and rightly be spent on the culture from which we have the most available resources in this country to study, or you will have to divide different groups of people up and give them each a different education and work very hard to get resources from elsewhere, and do an awful lot more overseas trips. If we don't want different groups of people receiving entirely different educations from the beginning of their schooling, then we will inevitably spend a lot of time studying the history of the elite, looking at the things paid for with the money of the elite (but not actually created by the elite), reading the poetry of the elite, reading books which require an understanding of Christianity, because like it or not, Europe was dominated by the "elite" and the Christian religion for centuries, so books written in English for centuries were dominated by references to that. We can't study everything in depth, but that doesn't mean we should study nothing in depth, or reject centuries of prior learning available to us, just because it isn't the prior learning of the culture from which we come (and which we have left in order to live elsewhere). And we will also spend time learning about the way the physical world works and how things are made, how to make things, how to explore things, how to analyse things, how to obtain employment and make your mark on the country in which you live, which happens to have a history we cannot change and cannot pretend didn't happen.