I never in my entire life heard of that happening to a milling machine. I have never even seen that happen on a drill press.
What exactly is wobbeling. If you take the tooling out of the spindle can you shake it from side to side with hand pressure?
Using your magnetic stand and dial indicator, put the probe end of the dial indicator in the spindle hole and rotate the spindle by hand. There should be almost no movement of the dial. Make the same check on the outside of the spindle. Check for any up and down movement. Put an R8 collet into the spindle and put an end mill or metal dowel pin into the collet hole and tighten the collet. See if it still wobbles with the dial indicator. If that is ok then try it with the drill chuck. Does it wobble with the tooling in the spindle but not wobble if the tooling is not in the spindle? Let me know the results.