Ended up chatting with Apple Support and later getting escalated to a call.
We ended up reaching two solutions (one still untested).
The first one (which is what I went for) was to create a new administrative account. In this new account the system could 'read' the system voice files; I even tried to get it stuck again by unchecking them and it still let me access the drop-down menu. At this point I just deleted the other account and signed in with iCloud to transfer all my files there.
The second option (which I didn't try) was to use disk utility. They linked me to this article to follow for future reference: How to repair a Mac disk with Disk Utility - Apple Support (CA). They told me to make sure to start from the bottom volume and work my way up running the first aid option.
If neither of these two options had worked, I would have had to reinstall macOS. How to reinstall macOS - Apple Support (CA).
Hopefully this can be helpful to someone.