Putting any bar in the chuck without first establishing if that bar is straight, or not, will not prove the chuck to be sound.
Further like all things in engineering the Chuck is made to a set of standards. Manufacturers give "Run-out" figures for their chucks and one of those is for a stated size bar, (usually ground, and not bright drawn), at a stated distance from the chuck jaws.
Thus putting a new chuck on a machine is not going to solve any run-out problems, unless a) the bar is known to be straight, and b) the test is not done to the manufacturers specification. Even then with a 3 jaw chuck the run-out is likely to be about 0.03mm or 0.001" about an 25mm or 1" from the jaws, unless the chuck is a Precision grade chuck.
Lastly the bar above has been machined in-situ, the lathe is doing what it is supposed to do. That is turn a true cylinder. This has no bearing on the accuracy of the chuck I am afraid.
Regards
Gray,