What Darkzero said
For a center a c'sink tool does not accurately follow the existing hole. You need a center to be accurate.
If the edge of the bore is wearing with a live center, then you are doing something wrong, there should be no movement between a live center and the work piece. If it is a large bore then at the time you are boring it you should cut the chamfer for your center at the same time, when you get to the bigger sizes not all centers are 60°.
For smaller sizes you should use a center drill, some times called a combination drill / c'sink. This will give an accurate location, the correct 60° angle and clearance for the tip of the center.
I have all the angles of c'sinks, 60°, 82°, 90°, 100°, and 120° but then I have been accumulating tools for a half century. Start by getting what you need for the job at hand, It is not a good way to spend to just get things for the sake of getting them, although getting a set of something is the best way to get things you do not have an imediate need for, Like a set of drill bits, I am sure that there are still a few drill bits that have never been used by me. I got a set of c'sinks and have used all of them at one time or another. Same with the set of center drills.