VGA port only works when cable is plugged in on startup

Question

Here at school we have a couple of older laptops running Windows 7.

The Win+P functionality doesn't work on those laptops. When you press this combination you see the spinning-wheel (loading icon) for a second and then it disappears. After that nothing seems to be happening. Same thing is true for the 'connect to projector' button.

So when I plug in an external monitor of a projector, it doesn't work.

However, when I plug in the monitor or the projector before booting up the laptop, the screen gets duplicated immediately. Note the external display or projector has to be turned on also, otherwise no luck.

Does anyone know what's going on here?