Your last paragraph seems misinformed.

https://en.m.wikipedia.org/wiki/Gymnasium_(Germany)

State-funded schools (a big majority) are tuition-free, as foreseen by the respective laws, even often on constitutional level. Segregation of students by parent wealth or income is looked down upon, to the point of being an exception to the constitutionally guaranteed freedom to have private schools (Article 7 section 4 of the German constitution, Sondierungsverbot). Of the private gymnasia, the vast majority is run by the Catholic Church on very low tuition fees (which is more easy as by Concordat, the Church receives a high percentage of the amount of money the State need not spend for a pupil in a Church-school); fees for schools who need to earn money by teaching are higher. Schools with fees generally offer scholarships.

In 2005, the German government spent €5,400 per student for those attending public gymnasium. This is less than what was spent on a student attending Hauptschule, but more than was spent on those attending Realschule.[22] Some Hauptschule and Gesamtschule students have special needs requiring extra help, so those schools cannot operate as cost-effectively as gymnasia.

I myself went to a public Gymnasium and can still remember that among my classmates we had a wide range re "parent income and status".

Literally from:

• 'single parent that barely gets by'
• 'teacher parents'
• attorneys / judges / cardiologist / engineers / etc.
• CEO parent of an international company

I literally:

• went through elementary school
• got a recommendation for going to a Gymnasium at end of 4th year based on my performance
• picked one out of the available Gymnasiums in my area (within what was doable by bus on a daily basis)
• then went there

And even if you go to "Hauptschule", that's not the end of education. One can still do what's called "M-Zweig", which gets you the equivalent degree of "Realschule". Then one can go to "FOS", which gets you a degree close to that of Gymnasium. It at least allows for going to a "university of applied sciences", which is less geared towards academics and more towards industry. Still can get you your masters degree or if a doctors degree if one really wants that and partners with an "academic university" (AFAIR).

Some children simply take that other route. Still, it comes down to ability of the child. Of course, having a non-supportive environment that doesn't believe in achild can make things harder on it. Some children might actually be motivated by "potentially escaping that".

But to say this education system is a "class filter" is just wrong.