Long story short, it doesn't seem as though Android is detecting the Google Backup Transport. The PA GApps package contains the GoogleBackupTransport.apk file, but I've also tried one from the Nexus 5 factory image (though I believe they're the same file).
In ADB, I ran the following command:
bmgr list transports
* com.google.android.backup/.BackupTransportService android/com.android.internal.backup.LocalTransport
My current setup is fitsnugly's nightly of CyanogenMod 11 and the latest PA 4.4 GApps (full, 2013-11-19). I've also tried with other GApps packages and other CM11 / AOSP builds with no success. I've tried both clean flashing and dirty flashing as well. Has anybody had any success with Google Backup / Restore while using a third party GApps package on CM11 (or any 4.4 ROM)? Alternatively, I'd appreciate any pointers related to where else to look or how to troubleshoot this, as I am not too familiar with this aspect of Android. I'm curious as to whether this is an issue with CyanogenMod 11, the PA GApps package, or [my] user error.
Thanks in advance!