Interesting! I know I helped someone over the phone and they had OS Mavericks. It asked to get an update (for iTunes), and then closed iTunes. They then unplugged the iPhone 8 plus which they had updated to the 13 (don't know if it was 13.1 or above) and iTunes recognized the phone.
They did not experience any of the "no phone showing" in iTunes. Of course they did not have HS and they were using a MBP.
But, clearly, there is an issue here for MANY users!! I have two Macs and I tried 3 iPhones!! iTunes would ask for permission, the dialog box would pop up on the iPhone and permission given, but the iPhone never showed up!
I was even on the phone with Apple for almost 3 hours and they had no problems just telling me that I need Mojave or greater, even after me telling them my hardware is not on their compatible list! Telling me that my choices were to buy new computers or use only the iCloud to backup my phones! (We even tried deleting everything and resetting the iPhone. That reset every thing, but it was still iOS 13! And, that did NOT work.)
That's not an acceptable solution Apple, especially since I am not the only one!
I still have an open case and I will be getting back in touch with them soon... I was just hoping to find a real solution.