Hello again, short post this time
I managed to correct the spindle accuracy by using a custom "holder". The holder was made using a piece of drilled out bar, and then hacksawed in half. The hacksawing was not accurate in any way, but as long as I assembled the two halves in the same direction, it had no problem holding the 1/4 inch rod in the 6 inch 4 jaw chuck.
And then I was able to get the runout to 0.01mm. After recutting the tapers, the spindle spun freely in the bearings!
Unfortunately, the gyroscope still has a vibration, caused by the fact that the 6 holes were not all drilled equidistant
only way to fix that is to remake the whole flywheel again.