Okay! So I've taken the advice of xgman - here's what I did:
a) Remove the backup drive from TM preferences (this will NOT remove backups)
b) Go to [root hard drive]/Library/Preferences, DELETE com.apple.TimeMachine.plist, and empty the trash. Restart your computer.
c) After restarting, re-setup Time Machine (as well as any exclusions you formerly had).
This step takes care of the "redirecting" to the Home Folder, but there are still plenty of weird issues in the UI. The first time I entered TM (the starfield view), it smoothly resized the folder and entered just fine, but subsequent times, it wasn't smooth (abrupt resizing of windows), BUT stayed in the same folder.
There was also a very weird issue when going back in time, sometimes the window view completely disappeared and went black. When I clicked "Cancel" (or pressed Escape), it wouldn't respond until I clicked again/pressed ESC again. Just VERY, VERY buggy and rough around the edges. Never had a problem with this in ML, and this backup IS from ML. I'm not sure if the same behavior would remain with new, Mavericks-only backups, though.
I'll share a video of what happens now when I'm finally able to upload it on YouTube.