You are on the right track.
Take the block and mark an "X" on both ends where the hole should be. It can be angled from end to end, BTW, if needed for a dropped handle butt.
Clamp a block of scrap wood in the drill press vise and drill down into it about 1" with a 1/4" drill bit. Cut a 1.5" long piece of 1/4" round rod and grind a point on one end. Put this in the hole, and check that it still lines up with the drill bit when the quill is lowered. Place the "X" mark on one end of the block on the point.
Now, drill a starter hole, about 1" deep, in the other end of the handle block where you marked it. Reverse the handle block, setting the hole you just drilled on the pin. You now drill the other end where marked. When the drill is in the block a bit over half way, turn the block around and finish drilling. The holes will always be in perfect alignment. You can use a similar process to drill a handle from the sides without the dreaded angled hole. It won't matter if the block is flat or rounded.
This procedure works especially well on stag tapers and crowns.
Stacy