even if the educational system is fixed, it doesn't change the fact that if you're poor you first have to find a job, and entry level jobs in careers are hard to find, so you might end up getting something you're overqualified for just to pay the bills, something unrelated, then maybe while you work at something unrelated life happens, you maybe get married, or have kids, and now leaving your underqualified but paying unrelated job becomes impossible, and by the time you can you've lost 10-15 years and your opportunities will be minimal.

If you're rich, you start working in your parents' business, you're also likely to intern in it, and/or in your parents' peers' companies, and if you're any decent you're likely going to be groomed for an exec position, a position the poor college grad will never be able to aspire to, and if you're not decent you're probably going to get some sort of face-saving "consultancy" position, paying again way more than the poor graduate can ever hope to get.

This is unrelated to the education system, and it's not something you can ever fix, because people will always try to make sure their family has an advantage, which means that as soon as a family reaches a position of influence it will use it to perpetuate its position.

Meritocracy is nice in theory, but humans as a species are wired a lot more for tribalism than for meritocracy, hence why to get some semblance of it you need a culture/government that fosters it, and those are in short supply.