Update! I decided to try Microsoft's tech support. Hey, I bought this thing from Microsoft, right?

They did the remote management thing on my machine, running some tool that tries to clean up any damaged files, and then since yesterday there was a new build of the OS (Win10 Pro Insider Preview), which wasn't available yesterday. So fine, downloaded, installed, and at least for now everything seems to be working. We'll see if this sticks around.

Fun fact: their remote management tool manages to restore itself after a reboot and pick up right where it left off. Neat.