If you do plan to use a centre drill as a countersink, bear in mind that the cone angle is 60 degrees, not the near 90 degrees more normal in woodwork for screw heads and the like.
Their 'proper' job is to sink the sink the holes into which a metalworking lathe centre seats, when turning between centres or when using tailstock support. The pilot part of the drill allows clearance for the centre's point, thus reducing the chances of damaging it and allowing the job to sit on the slightly larger diameter further up the cone. It also provides a small oil reservoir to keep the centre lubricated - not such a problem with a rotating centre, but sensible when using a dead (non-rotating) centre.
They're used in all manner of machine tools to start drills, too. Lathes, drilling machines, jig borers, milling machines doing jig drilling, and in cylindrical grinders and some inspection equipment, and so on. You can use them in a hand drill, but any wobble does tend to break the pilot off, especially if the operator is a bit 'cack-handed', so it's not really regarded as good practice - that's probably what happened to the OP's example. They break much more easily when hand-drilling hard materials such as steels than they do in wood.