Thanks for the tips and advise, Rog02! I actually had that chuck rebuild page opened in another tab on my web browser before I even read your link. (as well as a couple others...) But it's great to get some independent confirmation that I'm on the right track with my research, and I appreciate your tips..
I've been trying to remove the chuck "gently" (without damaging it, such as if I'm going to rebuild or reusre it) without much luck so far. I've gotten it off once before, but apparently I reinstalled it back on too good onto the taper this time, and it's not coming off too easily! I've got the threaded coller loose just having trouble rotating it hard enough to get it to press off the chuck.
I just got a great deal on a replacement chuck I found on ebay that is basically a "new old stock" jacobs superceeded replacement for the original chuck. It's part number 34-44C, replacing the original chuck which is the 633C. (The "C" stands for the threaded retaining collar which I think is a pretty sweet feature to have and one that I definitely wanted to keep.) I got the replacement chuck for about the same cost as I could have gotten just the replacement chuck jaw kit for (plus it comes with a replacment chuck key, which I could use because I just broke the little tip off of on the original one trying to remove the original chuck trying to use it instead of a spanner wrench

) so anyhow, should not too bad.
If worst comes to worst, I can just use a big pipe wrench to unscrew that threaded collar with more torque and drive the old chuck off of there now. I was thinking about buying or fabricating a spanner wrench to do it but, now that I have a new chuck coming, no need to "save" the original part anymore. Once I do get the original chuck off there, I'll be sure to verify the spindle taper does not have runout, just to confirm.
I figure the original chuck jaws are likely worn out, after reading it is not that uncommon of a problem (someone even considered chuck jaws a "consumable") and anyhow that sure does seem like what's going on with this drill press. I can see daylight between a straight machined rod clenched in the chuck and the very tips of two out of three chuck jaws. Also, the surfaces of the jaws are uneven widths and shapes between the three chuck jaws. Probably spun on something or maybe was just never manufactured right from the beginning.
Edit: also noticed in the post above, I said it's on a JT2 arbor, but it's actually a JT33. (Weldingweb has a short time period to allow post corrections, unfortunately, so this is the best I can do for correcting that info.)
PS - with as much runout as it has right now, I am getting better drilling performance for short holes out of my hand held drill! I am looking forward to seeing how it drills with the runout problem fixed.
Attached is a pic of the replacement chuck that is now on its way.