Collett closers and collets in the spindle bore are as accurate as the bearings in the headstock are and if you bothered to read my comment, it said... The more you add to a spindle, ie; a chuck or faceplate or any spindle nose mounted accessory, the sloppier the TIR becomes. TIR always depends on the rotating concentricity of whatever you add to the nose, always. Of course there are other factors involved like tool deflection, ambient temperature, temperature of the headstock bearings and if the machine is properly levelled as well.
When I produce my bushings out of drill rod for my bushing dies, I always chuck them in a collet but that is about the only time I use a collet. Most times a 4 or 6 jaw chuck is on the spindle nose (plate) as it's a D1 6 spindle.