Honestly, it’s not just capitalism. Education is anywhere from free to really cheap in Germany, and we still don’t get many people from poorer families into uni.
I see the main problem here as a sort of class divide between people with university degrees and people without. For example: if you work in a public library and don’t have a uni degree you will never get more money than salary level 9 (4k/mo) just having a degree and not doing any more/different work more or less instantly puts you on 12 or higher (6k+)
This I think understandably makes people without uni degrees kind of resentful of those who do have them. And if you grow up resenting a certain group of people you are much less likely to join them.
So, no. “Just” getting rid of the cost won’t magically get these people into higher education.