I used to build pool cues for a living. We would use a lathe to install the joint pin but...

https://www.cuestik.com/5/16-14-titanium-self-aligning-joint-pin/qtipin-14/

That's the link to buy the joint pins. If you get one that is 3/8-10, it is meant to screw directly into wood in the shaft without an insert. That will be easier on your end.

When you drill for both sides of the joint pin (the butt and shaft sides), have both the butt and shaft over sized. So if and when you probably miss the exact center on both, you'll be able to screw it together, then sand the joint together. Then you'll have a perfectly matched joint between the two.