Thanks for the details. Would it make sense to drill and install the bushing first, before final turning to ensure concentricity of the shaft?
Michael, that's definitely a good idea. I made another one today and tried a few different things, unfortunately didn't read your comment before hand. Here's a summary of my adventures:
Changed material to Texas Ebony, the hard material definitely added some challenges. The first attempt I drilled the hole for the bushing at 1/2" and enlarged using sandpaper. I then inserted the holder rod into the bushing. I then used my tailstock with a center to press it in. Unfortunately the handle cracked and the rod got stuck in the handle, total failure.
My second attempt with the Texas Ebony I drilled with the 5/8" like I did with the mesquite one. I used a similar procedure for pressing in, but put the holder rod 90 degrees offset so it is on the bushing but not completely inserted. This allowed me to remove the rod with very little force so I didn't risk gluing the rod in the handle (or pulling the bushing out), but I was more sure of the alignment. I still had to use some sandpaper on the hole because the holder rod wouldn't insert freely.
In hindsight, I probably would have had things much easier if I had glued in the bushing first. Also, with a softer material I probably could have drilled at 1/2" and chiseled the excess like originally planned. If done correctly, I think this would also help with centering and alignment of the bushing. Lastly, I'm thinking 2-part epoxy might be a better choice compared to gorilla glue. The gorilla glue foam out on the inside which was a hassle to clean out.