Just experienced the issue with iPhone 13 to iPhone 16. Chatted with Apple online.
Need to update old iPhone's iOS to latest version. So press X to stop the process on old iPhone. Go to Settings - General - Software Update.
On iPhone 16... I had to let it go to sleep, wake it, then hold power & volume up to force it off.
After old iPhone iOS is updated then...
Check the old iPhone still has some battery charge left. If not then I would charge it some.
Turn on new iPhone 16.
Follow instructions at support.apple.com/en-us/117383
Using a wired connection is better than wireless (and maybe faster), so go into your old iPhone box and find that USB-C to lightning cable to connect old iPhone to iPhone 16.
The whole process was done within 5 minutes or so.
The new iPhone 16 even connected to my home Wi-Fi automatically. All my apps stared downloading afterwards. Pretty cool when it does work.
Very frustrating to have waited over an hour when it was stuck. I still think Apple has a USER EXPERIENCE ERROR.