I think the thing that was preventing it from automatically backing up wirelessly was related to the encryption setting.

I had previously backed up the phone manually, wired, encrypted, many times, and also backed it up manually, wirelessly, encrypted several times. So I know that encrypted backups work. But it was refusing to automatically back up, wirelessly or wired.

After Tom confirmed that it *should* have worked, I suspected something related to the encryption. Steps I took:

I pressed the "change password" button for the backup encryption. My goal was to just force it to rewrite the password without changing the password, so I entered the same password. It gave me an error message saying that the password was invalid, which was weird (I was certain of the password).

So I unticked the encryption box. Again it prompted me for the encryption password to unlock the encryption. I entered the same password again. This time the password worked and the encryption feature became turned off.

Then it immediately began an automatic wireless backup. It's doing the backup now.

I want it to be encrypted, though, so I'll try re-activating the encryption again, and see if it continues to work after that.