I have had pieces start working out of the chuck jaws before, this was caused by the steady rest holding the piece out of line with the chuck jaws —- holding the piece to the side or up/down from in line with the chuck jaws.
If high or low, a tit will form above or below the cutter bit when facing the work. If sideways, a bit harder to see, however a small center drill will often break the end off it when trying to center drill. A small drill bit will often wiggle around trying to find center when there is no center drilling, a larger bit maybe seen pulling to the side.
As stated above, try to get an idea of where the center of the piece is, use a “V” head on your combo square, draw lines from a few sides, this should at least give an idea where center will be. A badly cut end just makes this exercise “more interesting”. Try to line up marked center with a center in the tail stock by moving the steady center feet/fingers.
Note a steady rest does not center stock, the centering part is up to you to establish by making a centering point in/on your work. When doing this would suggest turning as slow as possible, suggested cutting speeds do not apply here.
I hope this may help, and not confuse things.
A bit late to the show.