If the goal was to turn a pepper-mill then you're done. If the goal was to turn something that will last, that perhaps a great, great grandchild will pick up, look at the initials or name on the bottom, and ask "who was that", then you're not done. While I've never turned a pepper-mill, my guess is it has its own set of issues. One, of course, it's wood. Being hygroscopic, wood needs to be properly treated if longevity is of value.
Depending on wood species, I can burnish as good a finish as all the Jeff Jewetts - problem is, it's short term and there is zero protection. I can get a teeny bit more longevity with a bit of wax - the operative here is "teeny".
Next, I can finish with oil and get a great finish with considerable improvement in longevity - that can be improved with one of the many home-brews or store-boughts that combine oil / solvent / air-cure resin of some sort. These are generally "wipe-on" which meets the "git'er done fast" requirement.
After the wipe-ons, you evolve to lacquers and ulitmately the high performance/nasty stuff.
No doubt the person picking up your pepper-mile will see a beauty and pride of execution not experienced with a "store-bought" - if you used that magic log, perhaps the simple act of grinding pepper has been elevated to the sublime. I don't think it unreasonable that a person picking up that same pepper-mill ten or twenty-years from now shouldn't have the same experience - maybe more so.
If our spirit goes into our work, and our work lives on, isn't that immortality?