Why does restoring iPad not restore e-mail? Best way to fix? iPad Air 2 was stolen.

Six days later, iPad Pro arrived as replacement.

iTunes restore from local backup.  Makes me pick a new security code instead of restoring that.

Also, I open mail and NONE of the accounts I had before are there, but there are three (only) messages in a generic inbox from two of those accounts.  And they are three weeks old; lots of messages came in since.

Why is this, and is there an easy fix?
I am assuming you made a non-encrypted Backup on iTunes, since you had to make a new code.

The Apple KB article About backups for iOS devices states:


  From your Mac or PC, you can make a backup of your device in iTunes. Syncing your device with your computer isn't the same as making a backup. An iTunes backup includes nearly all of your device's data and settings. An iTunes backup doesn't include:
  
  
  Content from the iTunes and App Stores, or PDFs downloaded directly to iBooks (You can back up this content using Transfer Purchases in iTunes.)
  Content synced from iTunes, like imported MP3s or CDs, videos, books, and photos
  Photos already stored in the cloud, like My Photo Stream, and iCloud Photo Library
  Touch ID settings
  Apple Pay information and settings
  Activity, Health, and Keychain data (To back up this content, you'll need to use Encrypted Backup in iTunes.)
  


I marked the points you are probably concerned about (your personal executive summary) :)
I am sorry to say you have to add them again from my perspective.