I had the exact same problems as everyone is describing here, and I found the solution.
The problem appears to be related to how the initial installation/restore was done on the new phone, and almost nothing will fix it. The answer is:
1. Make sure the iOS version of your new phone is updated to the latest (or ideally matching the iOS version of your old phone).
2. Resent your phone and erase all content and settings (Settings -> General -> Reset -> Erase All Content and Settings
3. This will restart your phone as a new out-of-the-box phone, and after a few steps it will ask you if you want to restore your phone from a backup. Choose yes, and select your backup to restore from.
Then you'll see that everything will work perfectly. The only disappointment is when I discovered certain apps that are no longer available on the App Store will no longer work. For those apps, you'll be given the option to either delete them or keep them (but no use keeping an app that doesn't work).
FYI, I've just upgraded from an iPhone 6s Plus to an iPhone X. I have about 150 apps (including some that are no longer on the App Store).