I broached this issue to my coworkers. Turns out the system account owned all the folders in question, not allowing us to make changes to them via the install package, even though we were all local admin on the server. We had to take ownership of the folders in order to properly run the CU.

Something hinky happened to the permissions on that server several months ago. We're thinking about going back, revoking all the share / drive perms, then re-adding them one at a time and testing to ensure we get them set up correctly.