I think it’s about aspiration. Most people - regardless of class- tend to do what the people around them are doing.
so if all your friends go to uni, you will do it too.
Also, believing that anything is open to you. Never make your kids feel like they don’t belong anywhere. always tell them they can do anything they want if they work hard.
This all brings a natural confidence which is key.
It’s not a popular opinion but the state education system can provide everything a student needs to get into any uni they want if they put the work in.
im from a working class background myself, and in a profession which is full of privately educated, upper middle class. I never really worked THAT hard at school, but was in a group of academic friends ( not total squares though!), went to a good uni.
the only difference between me and my schoolmates who didn’t go to uni/ leave our home town, was that I wanted to do it and had confidence in myself. I thought I was the cat’s pyjamas!
also, I think the biggest difference is social skills. I hate to say it ( I’m wc background myself) but I find the mc parents at my son’s school far more chatty and sociable than the working class ones. Often once I’ve got to know them, they open up and are lovely, but it’s harder work and I’m always the one to start the conversation. I notice it too with school engagement etc. I think wc people are more likely to hang back, not put themselves or their views forward and so miss out on opportunities.
i think making sure your kids have hard working academic friends is also helpful.