If you see satellites then likely you see even more stars. Unlike satellites the stars barelly move (actually they do, see "proper motion" [1]) relatively to each other, so a catalogue of stars (two coordinates values and two proper motion values) along with the time of observation is sufficient to be used over decades, unlike NORAD orbit elements requiring regular updates. With stars you need just one image at a known time to find your location, with satellites it is much much more complicated: you need to know where the sun is, you need few images of a satellite or even a video (likely on top of image of stars anyway) to distinguish it from the stars and to solve the trajectory.
How do you find your location from one image of stars? It is possible if you have a precise vertical but you don't have a precise vertical on a moving UAV. That is, you need an inertial system on top that will provide you with a vertical.
With satellite images, you don't need anything apart from time. And no, you don't need to "make a video to see satellites move", you start with your approximate location, make an image and find satellites within a circle where each of them might be, starting with the slowest moving - furthest away from you - ones (they provide poorest precision of coordinates because parallax is small, but you need to start with something, but their search circle will also be smaller), locating those, you get better coordinates of yours and the search circle for each satellite becomes smaller, then you can find faster moving satellites too to get precise coordinates of yourself.