@jrich7970
If your lathe has an MT1 tailstock, you might have to look harder to find an arbor for the Jacobs chuck.
The reason I mentioned a J3 - MT2 arbor is because I know that is fairly common and I thought you had a MT2 tailstock, but did not know.
Many Jacobs chucks have a J3 taper fitting (but not all) so that explains my use of the J3. . You will have to check that to be sure. . Sometimes there is a marking on the Jacobs chuck itself.
But MT1 is quite uncommon. . I would think that the required arbor does exist, but finding it could take a bit of searching.
You may be better just continuing to drill as you currently do with the drill press since you have developed a good way to do that.
Also, the need for a headstock chuck ups the overall cost, so .... carry on as you have been is my advice at the moment.
But, YES, it would be better drilling on the lathe. . And easier. . The accuracy would be improved but that benefit is not such a big deal because any drilling inaccuracies are largely compensated for after you glue in the tube and round the blank.
My view is that I like to achieve good accuracy in every step of the process .... just my obsession I'll have to admit.
I hated drilling pen blanks on my small drill press because the hole, altho' straight, nearly always came out oversize and non-aligned with axis.
And I use a metal-working lathe for the whole process ... drilling, turning, trimming, etc. ... mainly because that was my prior skill set.
If you ever become committed to kitless pen-making then you may want to upgrade your lathe and go with a metal-working lathe.
As far as your turning activities other than pens, I cannot really comment. . I've made only pens (2000+) with some of them being kitless.
Making kitless pens often leads into doing some somewhat unusual operations for which a metal-working lathe is better suited.
Some of those operations can even involve working with metals !!