> I had to research what gymnasium meant, in the US that just means P.E. which is a class where you do generic sports for an hour each day.

Except... that's not it:

https://en.wikipedia.org/wiki/Gymnasium_(school)

> Are you saying, in Germany, you can't choose to go academic route in primary/secondary school? The teachers and school decide if you will go into vocational school for mechanics, electrician, etc?

There is an end-of-the-year test for kids in what would be the 8th grade here (and this is not Germany, but the Netherlands), if your last year's teacher sucked or simply doesn't like you then you're off to the vocational school or at best HAVO because there are a limited number of slots for VWO (Athenaeum / Gymnasium).

> That seems to imply class mobility is nearly zero in German education.

It's not zero, but it isn't nearly where it should be, and again, my experience is mostly with NL. Merit matters but it certainly isn't everything and there are certain schools where the gatekeeping is very visible. One way in which this happens is by keeping the number of slots for the highest level artificially low in spite of demand. You then have the choice of moving out of a region to a location where there is room or to accept a lower grade of education for your children. This is very frustrating, especially because kids with the right last names of course always mysteriously get in.

The better schools have a system where they share the first year of high school between all of the pupils and only then do they give the option to choose which track a particular pupil wants to follow. But these are not in the majority.