I solved it. I had to download and install DirectX End-User Runtime Web Installer from Microsoft: https://www.microsoft.com/en-us/download/details.aspx?id=35 and then reboot. New Oculus Home seems to work fine now.
@LZoltowski
I ran every single executable in your archive except the directx one because it wasn't digitally signed. I apparently already had all the redistributables except 2005 (people still use them?). The OVRRedistributableInstaller.exe and ocu…