Okay, video frames not being the same as individual photographs makes sense. Still, I'm pretty sure the FS700 has a setting for image stabilization for videobut I could be entirely wrong. As a part of the camera body as opposed to in the lens. I'm any case, it's good to know it's not commonly used in general.
Yeah, the 700 can shoot in 4K with something like an Odyssey 7Q, but not stock. Might be time to seriously consider the add-on.