With apologies to the 40 preceding posts regarding possible redundancy, here are my observations.
If there is runout in either the center drill or the setup, a center drill will orbit around the true spindle axis. If you push forcibly, it will start off the true spindle axis as play in the spindle allows. When the main cutting edge contacts the work , it will tend to run true and snap the tip. With the larger center drills, they just carve a new spot but smaller ones will break the tip. I was using 0000 and 00000 center drills with tips of .010" and .005" and breaking them regularly. A solution is to make light first contact and allow the center drill to find the true spindle axis. Once it has done so, as evidenced by the lack of wobble in the tip, you can proceed with the main cut.
If the broken tip leaves a stub, the tip can be reground to make a spotting drill
As to brand, all my larger center drills are house brand from Travers or MSC. with the exception of two Malcus No. 7's picked up at a hamfest. The 0000 and 00000 were purchased from Small Parts Inc. , no longer in business, and of unknown brand but likely to be German made.