My drill press quill has a 2" throw - in theory, that matches many (but not all) pen kits, but in practice, that's not really enough to align the bit and allow it to make a clean opening at the bottom of the hole.
My lathe has slightly more travel, but even that is limited, and there are kits that require even deeper holes.
So my take on this is that apart from spending a lot of money on specialized tooling, there is no simple answer.
When I'm drilling on the drill press, I use an approach similar to Ed's suggestion. But instead of bowl blanks, I keep a pile of scraps of 3/4" timber under the bench. I drill the hole as deep as it will go, lift the blank and slip a timber under the bottom, and then continue drilling. If necessary, I can add more timbers to get additional depth. And because the timber scraps are reasonably sized (4x6" or more), they will sit squarely on the drill press table.
On the lathe, I follow Dan's suggestion - I drill as deeply as I can, back the bit out to clear the swarf, but before advancing the quill, I move the tailstock until the bit bottoms out in the hole, and then slowly drive the ram to make the hole deeper.
Of course, another interpretation is that this merely illustrates that we never outgrow our need for newer and better tooling - - -