I don't think the PCB is that much smaller than the cooler. From the looks of images at least.
Do you really need the Strix if you're gonna replace the cooler anyway?
The only reason to get the Strix is for the cooler. Frequencies will be similar on all 1080 cards anyway. They all boost to about 2000MHz.
My 1080 Gaming X boosts to about 1976MHz in game mode.
Performance-wise, there won't be a huge difference.