In Germany there are, as far as I know, also a number of ways to get an exception. The normal way to attend Uni if you did not get the right Abitur from your high school however is to simply repeat high school from age 18 to ~21. This is called "Zweiter Bildungsweg" (Second Path to Education).

There is the colloquium doctum in the Netherlands, but you're still at a relative disadvantage if you're not put on the right track (VWO) as a child. So while the Dutch do a lot of things much better than the Germans, I think this one is only marginally improved.

Also cost of living in the Netherlands is far from cheap and as far as I know the Uni doesn't really help with that. So I'm still not sure how one would go about completing five or six years of full-time studies here without any financial assistance, either from their parents or by taking out a loan?

At the end of the day, it's always possible to go to Uni somehow, even if you had the bad luck to get sorted into the wrong bucket as a child. But it obviously takes much more effort than would have been required if you were placed in the right high school from the start. And even once you are admitted to Uni, not having a steady stream of passive income puts you at a huge disadvantage compared to most of the other students that do have it.

So my point is that, even in Europe, having wealthy and well-educated parents still puts you at a huge advantage when it comes to education. It's not exactly a solved problem here either.