Then I tried to center a hole in a 3" long cylinder, using the exact
same mechanism--and it didn't work. The hole was way off--about .05"
difference from one side to the other.

I started with a sherline...
I suspect the biggest source of error is that you have a long work
piece support by tiny jaws on the sherline 3 jaw chuck. ie you
have a 3 inch piece support by 1/4 inch jaws. This will not force
the work piece to be straight. In this manner the end of the work
piece probably has a lot of runout. If you can face the end of the
work piece (next to the chuck) so that it is square it will help
a lot. The faster you run the part, the more the spinning force
will encourage it to move off center. The thiner the piece, the
more it will bend. A steady rest will help a lot. Screw machine
length drill bits will help too. You might have better luck with
the 4 jaw. Do you have the key in the headstock with the headstock
preloaded by pushing the chuck as you tighten the screw? See the
sherline manual.

chuck