Why is my "sweep second hand" actually moving in small jumps?

That is the way is it supposed to work. The 'sweep' motion of a mechanical watch movement's second hand is not like the sweep of the second hand on an electric clock. Depending on the 'beat' rate of the mechanical watch movement, the second hand typically moves 5 or 8 times in a second. That is considered 'sweep' in a mechanical watch movement.

Due to the "persistence of vision," the limit of human perception is about 18 movements per second. That's why early movies from around 1900-1920 which were shot at 16 frames-per-second look jumpy, but modern film (24 frames/sec) and TV (30 unique frames/sec in the NTSC format used USA & Japan) give the appearance of continious motion.

So the 'sweep' action of 5 to 8 movements/second looks somewhat, but not perfectly smooth and continuous.