FWIW, get the lathe properly set up, and then if there is a problem start worrying about wear or slackness.
Your machine is a hobby lathe, and virtually unused, so unworn. It did nor cost anything like as much as an industrial toolroom lathe. So don’t expect the same level of precision from it.
Thousands of hobbyists are quite happy with the performance of their lathe, “out of the box”
Be realistic, and don’t expect a three jaw chuck to hold work concentric to much better than 0.003″ (0.075 mm)
We do not work in temperature and humidity controlled Standards or Calibration rooms, so our measurements are not REALLY accurate.
So no delusions absolute accuracy!
If you are tugging on a bar sticking out of the chuck, you are not reproducing the conditions under which the machine actually works. Once the machine is rotating, temperature and oil films will affect behaviour. Precision grinders are run for some time to warm up before starting work for real, for this very reason.
Get it set up properly, first of all.
Set it up as Ian Bradley advises in “The Amateur’s Workshop”, and “The Myford 7 Series Manual” (“Rollie’s Dad’s Method”) to remove twist from the bed.
Don’t try to run and complain that your shoes fall off, if you haven’t tied your shoe laces!
Then fit the 4 jaw and make an alignment bar. You will get useful practice clocking the test bar (Preferably precision ground, or silver steel) until it is concentric.
The accuracy with which you do this will determine the accuracy of your Alignment bar.
Then you can check, and if necessary, adjust the alignment of your Tailstock.
Once you have carried out the basics, you may be surprised at how well the machine performs!
Howard