Problem solved. I finally decided to compare the module list side-by-side and there actually was one missing. It turns out that there are two Windows Authentication modules:

On the server, the managed WindowsAuthentication module was there, but not the native WindowsAuthenticationModule highlighted above. Why it was configured that way is anyone's guess, but apparently if the native module is not loaded, the managed module will cheerfully load and silently fail.
So for any future readers who encounter this problem, make sure you have both modules loaded modules loaded, because IIS will notwill not warn you warn you if one of them is missing.