So in *my* shop, with *my* set of safety protocols, I always use gloves when using the lathe. No, not nitrile ones, but real, skin fitting kevlar gloves with very thin gripping surfaces. (the cut resistant variety)
--- now before you guys blast me about OSHA, all safety protocols are a complete system with compromises and assumed risks and mitigated risks. I have written exception protocols for OSHA Alberta, so I've had to write in these kinds of things...
IF:
1) you NEVER have your hands closer than 6" from any moving part, and keep your hands 12" away for normal operation
1a) Your lathe files are always used left hand and are more than 14" long in OAL
2) your shut off on the lathe cannot be restarted with one hand in one operation with the other hand near the rotating parts
2a) My lathe has a start button on the headstock and then needs the lever on the carriage to start the spindle, which is more than 24" from the chuck... (yes I suppose I could touch the chuck with my left hand, but that is usually on the carriage handle)
3) the glove is a close fitting one that does not have any extra 'floppy material than can get stuck in things, I.E. like a nitrile or stretch glove that needs hand-proximity to catch.
... only then can I see using gloves on my lathe, nitrile or not.