John, but have you tried that solution? In my case I had the same poroblem: while Illustrator was already running, and it was maximized, I dragged an artwork on its shortcut, and its window got small, even to a degree that the file couldn't be open. After I modified my shortcut (actually embedded in Total Commander that I love to use, although it shouldn't make a difference) to open the target app in maximized mode, the window stayed maximized.
The difference is that the method described by @emil emil and used by me uses shortcuts to open the file, and you are opening the files by double clicking. So the only advice I'm able to give is to create an AI shortcut in 'maximised' mode and use it to drag&drop your designs.
PS
Personally I never save artwork in AI, only in PDF, so my double-click action is to open the file in PDF viewer.