Success! At least I think so.
First off, in preparation to doing the actual work on the spindle taper:
I wanted to double check all the runout measurements on all surfaces to have a baseline as to what exactly I needed to do. Plus I wanted the Dremel tool mounted to the toolpost to be aligned to the spindle taper as closely as I could get it.
I worked on the lathe a day or two later, and I believe I now have satisfactory results. I was worried about the spread between the 3 jaw chuck's gap (0.002") and that of the 4 jaw chuck (0.008"). But I think I hit the sweet spot grinding the spindle taper for both of them. The 4 jaw chuck seems to fit flush when I crank down the camlocks using a socket driver instead of the key wrench. At least I can't get a 0.0015" feeler gauge between the surfaces. That is the thinnest material I have in order to try to do that. So I am hoping the surfaces are actually clamped together now, and there isn't something like a 0.0013" gap there. I can't see any daylight between the surfaces, but the outside rear lip edge of the chuck doesn't give me a real good angle to see any possible gap very well. Takes some solid whacks with the plastic headed mallet to break it free from the spindle. I didn't want to chance taking off any more metal, but I did polish the spindle taper surface a bit with a very fine grit after checking the fit with the chucks.
With the 3 jaw chuck, it will actually pop loose from the spindle when I loosen up the camlocks but there doesn't seem to be any lateral play at all that I can tell. At least when I put a milling bit in the jaws, the runout hasn't changed at all, even trying all three different positions for the camlocks, so repeatability seems to be OK. I tried different diameter bits in the jaws, and best I could get was around 0.0045" out of any of them. It was most often around 0.006" with most of the different diameter bits I tried. But in any event, I don't believe that runout is being caused by the chuck not fitting flush against the spindle face any longer.
The other two chucks I have remaining after sending others back to the sellers (ER40 and 5C collets) fit snugly and require modest persuasion with a mallet to break them free of the spindle taper. Runout for both of those chucks on the inside surface where the collets reside dramatically improved with that now being right around 0.0005". So that isn't bad at all. For me, anyway. While working on this, I spent considerable time working on that little bit of runout out on the spindle taper noted in the above video and I think that helped make a difference. All in all, I spent roughly 3 days working on this. So obviously I was taking my time.
So it appears I may have this problem behind me now. I am REAL glad I didn't have to replace the spindle. I think that would been a real PITA to do.
Anyway, just providing a followup if anyone is still interested. Thanks for all of the advice and help with this, but I decided working on ONE part (meaning the spindle) was better done than working on all of the chucks.