Okay, I was totally on the wood way with everything concerning Viewports.
Viewports render the whole scene, not only their children
The problem was that I had the camera as a child of the player kinematic body to be able to set its position relative to the player. When I replaced the Camera by a Viewport with a Camera, the relative spatial transformation got lost; I suspect because Viewport is not a Spatial node and that makes it lose relative positioning. This also explains why the tutorial video had to create dummy cameras under the car node and on every step update the real camera's (the one in the Viewport) transform with the global transform of the dummy camera.
Viewports as non-Spatial-nodes prevent relative positioning of child nodes to parent nodes of the Viewport.
Finding the source of my problem has taken me several days now. There really should be at least a hint in the Viewport doc about this. Better even: The editor could display a warning sign, if a hierarchy of Spatial nodes is interrupted by a non-Spatial Node because this loses the ability to position relatively to the parent Spatial.