The spindle chuck on the "china motor with flexible shaft" has a rather non accurate chuck. I can see the mill is not centered on its axis.

Is there a way to make this chuck more accurate, or are there good alternatives to this chuck that can be fitted?

Sure you can change it; but I'd be certain it's the chuck and not the spindle itself. I pulled one at random once that was like that, but I couldn't find the problem for the rest of that box, about 8 spindles. That's classic China for ya. (Also why I called those spindles a "sample order" hehe...) I believe they are using a "Jacobs Taper" for the chuck, which are sized as JT1, JT2, JT3. A chart of the sizes:

JACOBS TAPERS

JACOBS LARGE SMALL
TAPER# DIAMETER DIAMETER LENGTH
---------------------------------------
0 .25000" .22844 .43750
1 .38400 .33341 .65625
2 .55900 .48764 .87500
2 short .54880 .48764 .75000
3 .81100 .74610 1.21875
4 1.12400 1.03720 1.65625
5 1.41300 1.31611 1.87500
6 .67600 .62409 1.00000
33 .62401 .56051 1.00000
---------------------------------------

With the next iteration, all of the spindles will be checked for runout and the bad ones will be culled.