Runout on a ATDP16B trade drill press

I own a ATDP16B drill press. It performs very well except from the runout. You can actually see drill bits vibrate at the tip. The runout is not consistent as it differ every time I change bits. Sometimes its less, sometimes more. Even with the same bits. I’ve measured with a dial indicator to be 0,012 just below the jaws and it gets worse further down. The bits are fine. Runout on my handheld blue bosch is consistent and less (!)
0,007. The jaws are definitely a source to the low accurecy. I believe the taper is another. I,ve painted the arbor taper with a permanent marker. After inserting and removing it one can clearly see that the taper mates only at the very thickest part and not evenly. I understant that the chuck an the arbor are not the best quality. I want to upgrade to a better one, but first I need to know if I can expect the runout of the arbor socket to be low enough to take advantage of a quality chuck, say an Albrecht or Jacobs.

I presume that you have only recently purchased this machine and would suggest that you should be talking to Axminster Tools to see about an exchange/repair under your warranty. You may be able to upgrade at the same time - but may not actually need to do so.
Best wishes, Philip.