Are you using the correct screw?
Screw Countersink Angle
Most aircraft countersunk screws have a 100 degree countersink angle, whereas in the US most commercial grade countersunk screws have an 82 degree angle. England uses a 90 degree countersink angle. MIL-STD-1515 "Fastener Systems for Aerospace" requires 100 degree countersink.
The 100 degree angle is preferred when fastening into soft materials as distributes the pressure over a larger area than the 82 degree screw.
Another consideration is head height. If you notice the two screws above, the 100 degree has a thinner head than the 82 degree. If your sheet thickness is thinner than the head height of the flat-head screw, then the head will penetrate past the bottom of the sheet. This will require that you also make a small countersink in the tapped part that you are attaching to. It also means that the resulting clearance hole in the sheet will be larger than the recommended one. The 100 degree can be used in thinner sheet without the head penetrating through the skin. Note, there is such a thing as a "undercut" 82 degree flat head screw, although not common, it has a shorter head for use in thin materials.
Typically countersunk fasteners are designed so that they do not fracture at the head, so they should support the full force generated in the shank. The larger considerations are the mating parts and how they take the shear and compression stresses.
In a multi-fastener group, if there is any real out-of-position error in the alignment of bolts to the countersink, some of the fastener heads will see an added bending stress (the misaligned head is being forced to the center of the countersinksink). Contrast this to a more standard head, where mis-alignment of the bolt and hole can be accommodated, with no added bending moment.
When used with a nut, there is some float and alignment is less of an issue. When used in a tapped hole in a mating part, misalignment will add to the bending stress on the head.