Apple - How Do I Target an Airport Extreme Shared Disk for Time Machine?

Try using the following Terminal command then restarting System Preferences (if it's open):

defaults write com.apple.systempreferences TMShowUnsupportedNetworkVolumes 1

I knew it was going to end up being something simple that I was just overlooking...

As I Googled around more, some of the information I found contained images of the icons for the mounted drives. This turned out to be my biggest clue. Notice the drive as I had mounted it:

As it turns out, this is mounted and an SMB share. Easy enough to overlook, since all of my other network stuff is SMB. (I mainly use my Linux server for central file storage and share it with Mac and Windows clients, so SMB is the most universal.) However, Time Machine doesn't like SMB. I must have not even paid attention to the difference years ago when I first set it up. But by mounting it as an AFP share:

I was able to see it in Time Machine (even after reverting to show only supported targets) and am currently backing up to it.