I'm back with another update. I posted about my rebuild on OWWM and a guy suggested that I check the runout on my spindle because the 0.005" runout I measured before seemed high. So I did and the runout at 3 points on the spindle taper was in the ~0.0003" range. I'm estimating this by eye on my dial indicator that only has markings 0.001" increments and it moved less than half the distance between marks. I put the chuck back on and checked the runout with my drill rod again and got about 0.02"! I could actually see the runout in the drill rod. I rechucked the drill rod and still had about that range. Fast forward an hour, I'd removed and replaced the chuck multiple times but the runout was still there. I don't know what set of lucky conditions I had to get that 0.005" runout from before.
I thought that maybe something was inside the chuck that was causing one of the jaws to be out of position so I disassembled the chuck and cleaned it. On the bright side it operated much smoother but it didn't fix my problem. I then took it back apart and checked the runout on some surfaces inside. On the surface that has the printing on it, I consistently was getting about 0.008" of runout. I clocked the chuck on the spindle several times and was able to consistently locate the high and low points on the chuck. So, it seems that my chuck has a slight bow in it which strikes me as odd but it's what I'm left with.
I tried the ebay chuck and it was slightly better. The taper on the inside of this chuck is not as clean as the original one I had. But I tried to clean it and it felt pretty smooth. I figure the chances of getting two bent chucks is low but I'm not sure what else it could be. I've checked the tapers for anything that would cause an issue and cleaned them. When putting the original chuck on the spindle I snug the ring up,lightly tap it with a rubber mallet, and then snug the ring up again. The Ebay chuck doesn't have a ring so I tapped it in place with the rubber mallet. I'm pretty sure this is all there is to putting a chuck on a taper spindle.
I'm not sure how to separate my two issues (chuck mounted poorly vs bent chuck) because the only way I can check runout of the chuck is with it mounted on the spindle. If anyone has any ideas or recommendations I'm all ears.
As a reward for reading the depressing bit before I have pictures.
To get the chuck apart you press the outer sleeve off with the jaw midways retracted. Once the outer sleeve has been removed you can take the threaded ring halves out and remove the jaws. Each jaw must be put back in the hole you removed them from!! So keep track of them or figure out the differences.
Here is the chuck disassembled.
To help in assembly, Jacobs has marked the core of the chuck with numbers 1-3. Here is number 3.
Each of the jaws are slightly different as well. If you look on all three jaws at the location indicated you'll see they're all different. In this pic the bottom jaw is #1 and the top is #3.
If anyone has ideas let me know. I'm scratching my head here. It's late so there is a strong possibility I am overlooking something simple.