Nice gif! As John Quinton says, however: it would be possible to see something like that on a flat earth, if it were flipping over and over.
The problem this does create for FET is a little more nuanced:
It demonstrates that the stars are very, very distant compared to the size of the earth, because there is no visible change of perspective as the earth rotates (equivalently: star trails are circular and concentric).
But if stars are very, very distant compared to the size of the earth, and the earth is flat, then a star that is overhead in one place should also be more or less exactly overhead everywhere else at the same time. In particular, the pole star (which lies at almost the exact centre of star-trail photos in the northern latitudes) would have to appear at the same angle above the horizon everywhere on earth, which it manifestly does not.
The fact that equatorial mounts for telescopes a) do their job and b) need to be adjusted based on latitude is an equivalent proof. You point one axis of rotation at the pole star, and can then track any other star by aiming at it and then rotating the telescope around that one axis to correct for its apparent motion. This works anywhere on earth - but at different latitudes, the angle between 'vertical' and the axis of apparent circular rotation of the stars is different. Since the stars themselves can't be tipping over as you move around (other people would surely notice and comment on it), and the only other variable is the direction of 'vertical', it follows that 'vertical' is changing as you move around the earth.