I'm confused. Was the OP asking a question about working stainless steel?
Post #3 suggested that the OP hang on to the drill chuck to avoid a problem if the workpiece grabbed and the chuck came out of the tailstock. That's the origin of some of this discussion.
Couple things:
1) smack the taper together so it doesn't come out
2) no machine shop I know or have heard about espouses or allows anyone to hold on to the drill chuck in order to prevent a problem. Why? It's bad practice.
The good practice is to smack the taper together, know your materials, speeds and feeds, coolant, lube or not, depth of cut, etc., and to operate accordingly.
I know that a lot of lathes and power tools used by members of this forum are light duty and may or may not be overpowered by muscle but that doesn't make it good practice just because you can. A sharp edge powered by a thousandth of a horsepower will draw significant blood and part you from your measly pound of flesh.
When was the last time your drill press chuck came out of its taper? Doesn't happen often even when the drill grabs. It's because it's smacked in there. I know a lathe is a different beast and the moving piece vs stationary piece is reversed but if the taper slips, that's detectable immediately under manual operation and the red button should be handy.
More fingers than workpieces will be saved by good practice.
All right --- done. Thanks for reading and happy turning.
Peace,
Rich