I’ve found as long as you select the window, use the keyboard shortcut to select move as you noted, use the arrow keys to move it a few steps, after that move the mouse without clicking and the window often moves with it at that point.
But yeah it’s the side effect of an inbuilt setting to remember last location on screen when the app was last used. Just not taking into account that the specific screen location is no longer available.
When it’s only one or two a month it’s manageable, but now everything worth accessing is split over a dozen services. I gave the legal option a go and it became excessively expensive, so back to piracy. Both cheaper and more convenient.