I notice your machine has a chuck "made in England" so not "all american" I guess.
Do agree there are many chinese rubbish machines out there, some are better than others, mostly the ones from Taiwan.
Not possible to get a "true running part" from a 3 jaw chuck after it has been removed and refitted after machining outer surface. Any machinist will tell you that, it is one reason why there are 4 jaw chucks. 1 thou runout doubles when you bore the centre hole.
Most 3 jaw from England will have around 1 thou so it is not surprising unless the jaws have been stretched or the chuck mis-handled at some time in it's life.
You can get a cheap 4 jaw made in India for example that will give 100% accuracy for these type of jobs even though they may appear a bit junky. Suffice to say the tailstock centre is properly aligned to begin with.