How to fix iPhone drivers manually
Install iTunes from Microsoft store
I installed the latest version of itunes (Dec 2018 - 188.8.131.52) from the Microsoft-Store. This was the first time I did that. All previous updates were either direct downloads or downloads from within iTunes itself.
No more drivers?!
Previous iTunes installers had uninstalled older versions and installed both the device drivers and iTunes. The version from the Microsoft-Store app removed the old iTunes and it's drivers, but did not install new device drivers.
This error message came up:
Windows-Update vs Device-Manager
Although the error message in iTunes suggests trying windows-update, it really is the device-manager that does the trick. You can call it directly from the command line with: devmgmt.msc
And sure enough, there is an Apple iPhone listed there. But that is not sufficient. Right-click the iPhone and click "update drivers". Let the PC look for drivers on the internet.
Back to iTunes
iTunes now fully recognizes the iPhone / iPad
It is not quite clear to me why the installer from the Microsoft-Store behaves that way, but anyway: