Hang about gents. I had EXACTLY the issue that John mentioned on my Myford spindle. A very small burr was raised and I couldn’t find the error but faceplates etc all showed a definite but small wobble.
Sent hte spindle back to Myfords and the estimable Mr Parker, now I believe retired, diagnosed and fixed. Cost nothing but the postage. (typical Myford IMO)
So I would check the thread given the history. Get a facepalte or something relatively unworn and run it under a DTI. I would then face something, clock with a DTI, turn trhough 90 deg and reclock.
Then I’d look at the chuck, as described above.
A purpely personal opinion – I have on the Chinaman an ER32 collet chuck. Had to be backplate mounted. That is spot on. For the Myford I have an ER25 screw on collet chuck and that does show a tiny bit of run out. However, I have a Grip tru so for absolutely critical work, I use that. I should fix it, I know.
If I were doing it again, I’d do a proper backplate mount, and I’d bore and thread the backplate myself.