This is generally correct and a huge problem. Note that the details of the process differ quite a bit depending on federal state and decade. What may differ are

This is generally correct and a huge problem. Note that the details of the process differ quite a bit depending on federal state and decade. What may differ are things like duration of Gymnasium, whether recommendations are binding, whether there's a clear distinction between Hauptschule and Realschule, duration of primary school (e.g. 4 or 6 years), how easy it is to switch etc. In my case, primary school was 4 years, recommendations were binding (!), so you basically got put on a rigid track based on one person's opinion when you were 10 years old. Unsurprisingly, that leads to material biases based on parents' education and origin.