Why are longer sheet metal screws snapping on installation?
Torsion strain exceeding screw strength. It's likely that 3" of wood is creating some friction, even though your pilot hole is nominally larger in diameter. You might try a 3/16"ø pilot hole and see if that reduces the amount of torque needed. Deep holes of narrow diameters in soft materials are never perfectly straight so the extra room of a wider pilot hole is a fast and cheap solution. I remember being warned early in my career to pay attention to the screw head type and do not expect the same results when you use a screw gun on a screwhead designed to be turned by hand (as traditional panhead and button head slotted screwheads are). You might also trying lubricating the screws to reduce the torsion load. Beeswax being typical and safe here for both wood and metals. You could expect that even though some heads didn't snap off, that they may do so in service - being under the same stress. You can also use a higher strength screw preferably with a head designed to be power driven.