Not everywhere. Schooling is totally state-dependant.

And maybe the fiercest and bitterest war waged in state politics over the last decades were about school systems.

In Hamburg there are mostly comprehensive schools. In the largest state, Northrhine-Westphalia, there are still those separate tiers, but comprehensive schools are probably the majority.

it is not a class system, it's integrating some aspect of meritocracy. It is performance, not wallet based.

Performance is highly correlated with household income for fairly obvious reasons, regardless of system.

Of course, but there's a difference between income being relevant yet incidental, and embracing it as a completely appropriate model to relish in.