Additionally, we aren't limited to a flat surface or a stationary one, and it doesn't have to be huge, either. It only need block the sun, not the whole sky.Keep in mind what the shade panels are made of will make a huge difference in whether how much of the signal will be blocked, if any. The front of the dish has some type of plastic covering the transceivers. That material or a similar material could likely be used to shade the sun while having minimal impact on the signal.
I've seen covers for DirecTV and DISH dishes that were intended to keep snow from accumulating on the dish, while allowing the signals to get through. Those are different frequencies, so it just a matter of finding the right material for the frequencies that StarLink uses.