You're right, they aren't running it by themselves, but they are running it.

Particularly in the German system, I don't know as much about the Canadian one, the ministers are pretty autonomous. Expert underlings don't help if the top person insists on going in a bad direction. And parliamentary systems, particularly the heavily negotiated coalition governments we often see in Europe seem more likely to give out assignments based on power sharing criteria rather than their suitability. Sometimes you get lucky, like Baerbock and Habeck, and sometimes you get Lambrecht.