As a simple minded small lathe user I would query using grease in the chuck due to greases ability to hold onto swarf.
I know that chips will stick to oil but have more chance, perhaps, of being flung out when the chuck is rotating.
My small chuck has an oil hole not a grease nipple.
After I've oiled my chuck I wrap it in kitchen paper towel, held on with an elastic band, and run it at high speed. This gets the excess oil out quite well.