As you can see, there's more than one way to skin a cat!

Or a less macabre way to put it, more than one way to drill a hole.
The least precise version would be .... to grab a hand-cranked or power drill and drill the blank freehand ... Is it POSSIBLE to drill a straight hole? sure .... might be a little wavy or wobbly inside for the first few centimeters as you straighten your bit to go somewhat down the center. Will it be straight down the center of the blank? probably not.
The next step up would be to use a holding fixture for your power drill, such as a portable drill press attachment ... or to step up to using an actual drill press. This method CAN drill a very straight hole, and IF YOU LINE IT UP PROPERLY 90 degrees to your table and use a proper clamp/vice setup to hold your blank, you CAN POSSIBLY drill right down the center of the blank. The problem is ... is your table properly set up that way? Does your drill travel exactly 90 degrees in direction from the plane of the table? Is your material properly clamped?
The next step up from there would be the lathe. Why is the lathe more accurate? If your blank is spinning on center of axis WHERE YOU WANT THE HOLE TO GO and your bit is positioned properly, you will get a dead center hole right down the middle of the blank ... if your bit does not BEND. To keep it from bending, you can drill a starter hole and then drill with the full diameter bore rather than using a small thin pilot hole bit.
Another measure to keep the bit from deflecting or wandering is to make sure it is SHARP! (use a drill doctor!)
Any method that makes use of putting the drill bit in the headstock side and drilling their blank on the tailstock will TYPICALLY FAIL unless your blank simply CAN NOT BE HELD ON CENTER. (antler, horn, bone, out of round material that you just can"t center up on your lathe's clamping fixtures). For things like this, a drill press or a pair of vice grips can be used (or both together), but you CAN use your lathe for this operation with the vice grips.
Mark and center drill (any way you choose) both ends of the antler round. You'll use this center drill mark with your tailstock to try to keep things properly centered during drilling. Use a 60 degree center (doesn't need to be a live center) in the tailstock, or something similar that wont mar your work piece. Use of a draw bar with the drill chuck mounted in the headstock is probably a requirement.
Hold the piece of antler with one side centered on the tailstock center and USE THE TAILSTOCK WHEEL to push the antler up and onto the bit for your drilling operation .... drill slowly, holding the antler with your vice grips. Once you are about halfway through the blank, completely back it out and drill the rest of the hole from the other side .... this ensures that even if you are a TINY bit off where the two holes meet, each end will have it's hole perfectly centered. Make sure your bit can pass completely through, and use a good gap filling glue inside (as antler is likely to have interior voids to fill anyways, from the marrow pockets).
So ... that's a lot of ways to drill! We didn't even cover reaming, broaching, milling, or using an offset cutter to enlarge an existing hole!
In short ... there's a lot of ways to make a hole. Find one that you are comfortable with and gives you the results you desire. No one person is always right all the time, because it's only human to make mistakes. (don't forget to learn from them, though, ok??

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