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If the spindle nut is loose, then the rotor is loose. I had this same problem after I repacked the bearings. (Chilton said to back off a half of a turn which was to much)
The off balanced tires makes more since.
General rule for wheel bearings with a repacking would be to spin by hand the rotor while tightening the nut to about 20-25ftlbs. Once done loosen the retaining nut 1/2 a turn, and then retighten to a torque that's best described as "finger tight" LOL. (it's a very low torque value, at the shop we just did finger tight) and then put the notched out nut cap on it, and put the cotter key back in place.
If all anyone does on a standard spindle is tighten it down, and loosen it 1/2 a turn? And fails to tighten it down finger tight afterwards? That would explain the wheel bearings being to loose.
S-
(it's in the haynes manual on my desk, if anyone wants the exact value of 'finger tight' LOL)