First, look at your faceplate. Not all are equal. Back when some of us started turning they were bored for the standard wood screw, which meant tapered for a thicker shank, and a countersunk for a flat head. Now you just don't know.
What you'd like to do is have a screw suited to the faceplate. If they're innocent of countersinking, pan head sheet-metal screws might be your choice. They're not brittle, the threads are fairly deep, and you get a bit of advantage by getting the threads bearing on the sides of the faceplate bore, instead of putting a narrow shank through a wide hole. That's a recipe for flop.
If there is a countersink, go with oval head to get the best contact with the faceplate, as well as the benefit of centering that the countersink gives you.
Last thing to suggest, if you don't have a soft-start capability on the lathe, is to start rotating by hand to get a bit of the startup inertia taken care of. Else, after the first couple starts and stops, go back and snug the loose ones. I use a pin chuck, of course, and find this a useful procedure on heavy blanks of soft wood.