You have a lot of interfaces in your set up - the spindle Morse taper, the chuck Morse taper, the chuck's internal taper, the collet external taper, the nut and the test rod. Add to that the need for adequate torque on the nut before you can check run out on the rod. This is collectively called stacking tolerances but each of these can contribute some/most/all of the run out and you need to nail down exactly which it is or what each one contributes in order to figure out what to do about it. I would not grind/bore anything until you do this. As Tozguy said, make a diagnosis first.
I can tell you that a cheap collet can add 0.0007" of run out all by itself. A cheap Chinese nut can double that. This all matters if you are doing second operations work and need precision. For first operations work, it matters very little.