Used Bison 6 1/4" for $600 is way high, considering what you are going to put it on. One of those things that makes no business sense, even though you are not a business.
1) You can get 3 jaw chucks (if you MUST have one) new for quite a bit less, and they will probably serve you well. Check the importers. (I would not do that yet)
2) You can look that one over and see if there is an issue with it. I'd do that first, since it may save you a bunch of money, and at worst you will learn something..
I see two things in your videos.
A) I see the OD is slightly "out", as in 3 or so thou total. You made a great move checking that! Most do not think of it.
Almost any chuck from a halfway credible maker will be well under a thou total when checked. So, with that amount on the OD, it is dollars to dog poop that the mounting plate is "out" a bit (not made well), OR that there is something holding the chuck cocked vs the spindle centerline, etc. The chuck itself will be better than what we see there.
B) I see a whole lot of runout on stock gripped in the chuck. Too much for regular 3 jaw nonsense, and too little for jaws in the wrong places/wrong order. Crud in between stuff can do that, and so can a backplate that is "out" a bit, not made well.
What I would do is to unscrew the chuck, and look at all the threads and mating surfaces, looking for dings, dirt, chips, etc that can hold the chuck out of square. Check the back surface of the plate where it bottoms on the spindle, and the threads inside the adapter plate as well as the spindle threads.
If no joy, I would take the chuck off the backplate (there should usually be 3 or 4 screws going through from the front), and check that for crap in between. Also check to see whether the chuck is snugly mounted, or if it can slide around a small amount. It should be pretty darn snug.
If no joy, I would then re-mount the plate, without the chuck. Check the face of the plate at the outside, where the chuck mates to the plate. First, I would check to see if that surface is wobbling, using indicator with the plunger of the indicator pointing along the bed parallel to the spindle. It should be dead steady.
Next I would (if it is possible with the tip of that indicator) check the "spigot" of the back plate, the OD that fits inside the back of the chuck to align it, with the indicator radial as you did with the chuck. It should be dead steady.
If any of those things are not correct, the chuck won't be mounted right relative to the spindle. The good news is that fixing those is an easy turning job, and of course crap stuck in between, that you just clean out.
If none of those things seems to be the issue, there are "more interesting things to check as well.
Final question: Are the jaws snug and held steady even when not tightened on anything? Or do they wiggle around and rock back and forth a bit?