THE OBVIOUS-NOT-OBVIOUS FIX: Install latest drivers and patches from Windows Update
This was a company computer, and we only have access to very specific updates on-network, probably distributed via Active Directory by WSUS. This laptop usually functions like a desktop, hooked up to dual monitors, so I never move it, but I took it home with me and connected it to my home network and installed all of the latest patches and drivers. After that, I had no issues with RDP.
For those of you who found this question/answer who ARE running Windows Insider Preview:
I also had another computer experiencing this same issue that was running the latest version of Windows Insider Preview (not sure which build), and RDP had broken on that computer as well (unrelated to initial computer/issue/question).
I ended up having to reinstall stock Windows 10 in order to resolve that particular issue, because apparently the Insider Preview build had broken that functionality. Heads up -- when going back to stock Windows 10 from Insider Preview, you don't get to keep your files, so make sure you're backed up. Some of you may say that was a silly thing to do, and no in that particular case, that was not finding the root cause (but this wasn't the initial question). However, that computer was also very important for VPN access from home to work, so I chose the swiftest method of attack, as I knew I would need to RDP to and from that box in the very near future.