I took the spindle off the machine and put it in a vice to feel the runout properly. It's weird, it seems to be there at one angle and not another, then it disappears, then it comes back.

I could resist taking it apart, and it was easy. At the collet end a cap is held by four screws. The cap has 6 springs acting on a plate, then some steel shim washers, then a plastic washer and onto the outer housing of a single bearing. The shaft pulls out easily and there is a smaller bearing pushed onto the other end of the shaft.

I don't know about high speed spindles, or any other rotating machinery for that matter, but it seems like a reasonable design.

I'm wondering now if the play I feel is actually play at the other end of the shaft, the other bearing might be a loose fit in it's bore. It certainly didn't give me any trouble sliding the assembly back together but it could be designed with a taper to do that. The play does feel somewhat 'remote' and there are only 2 places it can be.