It also backed up successfully as soon as I turned the encryption back on and re-entered the password. However I ran into a snag:

When turning the encryption back on and re-entering the password, it gave me an error message saying that it was unable to set the password, and offering to try again. Trying again also did not work.

Then I turned the phone's LCD screen back on (it had turned off the screen because of a small amount of idle time). As soon as I turned the LCD screen back on, it was able to set the password.

As soon as it successfully set the password, then it began backing up immediately.

I am still not sure if these last two successful backups were "automatic". since they were in response to turning the encryption on and off. Both happened immediately after making a successful change to the encryption setting.

I'll have to wait and see if this tweak actually results in a truly automatic wireless backup.