Drills ground with four facets hardly need centring.
But you are unlikely to be able to grind four facets, to drill accurately, freehand. Even two requires practice and skill, particularly on smaller sizes.
A sharp centre drill should have a chisel point rather like a four facet drill.
Since a centre drill is not required to produce a hole of exact size, a small difference in lip length should not matter too much. The important thing is that it is sharp and cuts!
Maybe, in addition to ensuring that the centre drill is sharp, use a smaller centre drill, so that the tapping drill locates on the cone rather than the bottom of the blind hole.
Howard