That extra center hole is part of the manufacturing process. If you look at the tools used by horn workers, you will see examples of their button "drill". It resembles a modern Spade or Fosner (sp) bit. It has a center point to keep it lined up, and one or two outside cutters - only the old horn button bit does not have that wide "chisel" type blade used to drill a hole in wood. It just has a marking point on one edge and a cutting point on the other outside edge. So you flatten your horn, place the drill bit where you want to cut out a button, and drill. That center point holds it in place. Those outside edge markers/cutters then cut/scrape done a circular hole - like with a modern hole saw. Those two outside points can be either just straight to scratch through the horn, or could be filed into a better "cutter" shape.
C. Keith Wilbur's book The Picture Book of the Continental Soldier shows a pretty good drawing of the drill bit and the process on page 15, and talks about horn and bone buttons. But it also shows them then being covered with cloth. Page 16 shows casting pewter buttons.
If you have an old Spade bit, just carefully grind back that "chisel" edge leaving those outside cutter points and that middle point. After you have cut out your disks, you can drill however many holes you wish, or go all the way to covering them with cloth fabric.
There are lots of ways to cut disks out of horn or bone. That special drill bit just helps speed up the process, and keep them more even. It also explains that center 5th hole.
And the pressure you apply downward should take care of some of that possible ... chatter. A couple clamps holding your flat horn piece to a board will also help. You could even "nail" the horn to that scrap board if it helps - just in the "waste" areas between futur buttons.
Just my humble thoughts to share. Take them as such.
Mikey - that grumpy ol' German blacksmith out in the Hinterlands
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