This method of reinstalling iTunes worked to solve iPhone no longer appearing in iTunes.
I was having the exact same issue, with all up-to-date versions below, after the iOS 12 update, the iPhone no longer showed up in iTunes. Even when through a chat with Apple on the matter – they of course had no good ideas on a remedy.
To note, the "An update is required to connect to iPhone..." message appeared rather early in the iTunes install process. Clicked on "OK", so the iTunes install, and software update were sort of going at the same time. The iTunes install finished first but the software update was still going. It's shown as a tiny window with progress bar which tends to fall behind other windows. So I when I tried to open iTunes, it said is was being updated. I closed everything else, then saw the tiny software update window. After everything was done, the iPhone was back in iTunes.
Also, all my play lists and music were as before. So, an easy fix.
iTunes version: 10.8.0.150
iOS version: 12
macOS version: 10.13.6
Surprised Apple would release and iOS update and then we have to figure out how to get things working again. Sigh.