Can I just rub oil into the bare spots? What about partial nitro finish?
Yes, you can reapply oil or nitro back on the bare spots. Compatibility with the old finish could be an issue, I've never encountered it myself, but it happens.
How you want the neck to look will also determine how much work is needed.
I reshape a lot of my necks and I can make mine look either almost seamless or close enough. My necks are played in and my bodies are far from mint, so close enough is usually what I'm happy with.
This in my '02 57RI neck, love it but it wasn't "V" enough and to my hand it feels like it's tapering to a "C", so I changed it to my liking and since it didn't need to look new so light tinting and 3 passes with spray lacquer, which works for me but I know the downside of that, and I'm happy. I used StewMac and masked off every other inch of neck, on this neck it's obvious where I removed and replaced nitro to give you a idea.
Cosmetically it only need to look something like this.