Actually, on a 6", TIR of 0.003" isn't too far off of the mark. According to a factory partial drawing that we have on the Atlas M6-81B Bearing, in between the two Timken numbers is written "CLASS 2 FIT". According to ANSI-ABMA 19.2. the allowable "Assembled Bearing Radial Runout" is 0.0015". If a PO replaced the spindle bearings with the cheapest things he could find (which would be normal for too many users) he probably got bearings that don't even meet Class 4 (0.0020") made you-know-where.
Also, according to one of the catalogs, the three-jaw chucks sold by Atlas and Sears would only meet 0.0050". And the way in which the catalog ad was written didn't make it clear whether they meant 0.0050" TIR or +/- 0.0050", which would be .010" TIR.. So I would concentrate on getting the 3-jaw runout on an accurately ground gauge bar within that and for anything that needs to be better, use a 4-jaw. And concentrate on the other issues.