If you’ve seen the “A software update is required to connect to your iOS device” message, and even though you try to allow it to update it just sits at the beginning and doesn’t make progress, I may have an answer!
I first tried the usual things… restarting the phone, restarting the computer, verifying the App Store had no updates available, updated the OS on the phone, updated all of the apps, etc. No joy!
Then I found the solution. I saw a potential solution mentioned online that it might help to install Apple’s Xcode from the App Store. I had already installed Xcode since I use some of it’s tools already, but I had actually never opened Xcode itself. I launched Xcode, accepted the Terms of Services and other popups… then tried the phone again and the update proceeded as expected!